Date: March 28th, 2012
This is a collection of examples showing how the GAP system [GAP21] can be used to compute information about the probabilistic generation of finite almost simple groups. It includes all examples that were needed for the computational results in [BGK08].
The purpose of this writeup is twofold. On the one hand, the computations are documented this way. On the other hand, the GAP code shown for the examples can be used as test input for automatic checking of the data and the functions used.
A first version of this document, which was based on GAP 4.4.10, had been accessible in the web since April 2006 and is available in the arXiv (no. 0710.3267) since October 2007. The differences between that document and the current version are as follows.
The format of the GAP output was adjusted to the changed behaviour of GAP until version 4.10. This affects mainly the way how GAP records are printed.
Several computations are now easier because more character tables of almost simple groups and maximal subgroups of such groups are available in the GAP Character Table Library. (The more involved computations from the original version have been kept in the file.)
The computation of all conjugacy classes of a subgroup of \(PΩ^+(12,3)\) has been replaced by the computation of the conjugacy classes of elements of prime order in this subgroup.
The irreducible element chosen in the simple group \(PΩ^-(10,3)\) has order \(61\) not \(122\).
The main purpose of this note is to document the GAP computations that were carried out in order to obtain the computational results in [BGK08]. Table I lists the simple groups among these examples. The first column gives the group names, the second and third columns contain a plus sign \(+\) or a minus sign \(-\), depending on whether the quantities \(σ(G,s)\) and \(P(G,s)\), respectively, are less than \(1/3\). The fourth column lists the orders of elements \(s\) which either prove the \(+\) signs or cover most of the cases for proving these signs. The fifth column lists the sections in this note where the example is treated. The rows of the table are ordered alphabetically w.r.t. the group names.
In order to keep this note self-contained, we first describe the theory needed, in Section 11.2. The translation of the relevant formulae into GAP functions can be found in Section 11.3. Then Section 11.4 describes the computations that only require (ordinary) character tables in the GAP Character Table Library [Bre24]. Computations using also the groups are shown in Section 11.5. In each of the last two sections, the examples are ordered alphabetically w.r.t. the names of the simple groups.
| \(G\) | \(σ < \frac{1}{3}\) | \(P < \frac{1}{3}\) | \(|s|\) | see |
| \(A_5\) | \(-\) | \(-\) | \(5\) | 11.5-2 |
| \(A_6\) | \(-\) | \(-\) | \(4\) | 11.5-3 |
| \(A_7\) | \(-\) | \(-\) | \(7\) | 11.5-4 |
| \(A_8\) | \(+\) | \(15\) | 11.4-3, 11.5-5 | |
| \(A_9\) | \(+\) | \(9\) | 11.4-3, 11.5-1 | |
| \(A_{11}\) | \(+\) | \(11\) | 11.4-3, 11.5-1 | |
| \(A_{13}\) | \(+\) | \(13\) | 11.4-3, 11.5-1 | |
| \(A_{15}\) | \(+\) | \(15\) | 11.5-1 | |
| \(A_{17}\) | \(+\) | \(17\) | 11.5-1 | |
| \(A_{19}\) | \(+\) | \(19\) | 11.5-1 | |
| \(A_{21}\) | \(+\) | \(21\) | 11.5-1 | |
| \(A_{23}\) | \(+\) | \(23\) | 11.5-1 | |
| \(L_3(2)\) | \(+\) | \(7\) | 11.4-3, 11.4-4, | |
| 11.5-5, 11.5-8 | ||||
| \(L_3(3)\) | \(+\) | \(13\) | 11.4-3, 11.4-4, | |
| 11.5-5 | ||||
| \(L_3(4)\) | \(+\) | \(7\) | 11.4-3, 11.4-4 | |
| \(L_4(3)\) | \(+\) | \(20\) | 11.4-3, 11.5-5 | |
| \(L_4(4)\) | \(+\) | \(85\) | 11.5-5 | |
| \(L_6(2)\) | \(+\) | \(63\) | 11.5-5 | |
| \(L_6(3)\) | \(+\) | \(182\) | 11.5-5 | |
| \(L_6(4)\) | \(+\) | \(455\) | 11.5-5 | |
| \(L_6(5)\) | \(+\) | \(1953\) | 11.5-5 | |
| \(L_8(2)\) | \(+\) | \(255\) | 11.5-5 | |
| \(L_{10}(2)\) | \(+\) | \(1023\) | 11.5-5 | |
| \(M_{11}\) | \(-\) | \(-\) | \(11\) | 11.5-9 |
| \(M_{12}\) | \(-\) | \(+\) | \(10\) | 11.4-2, 11.5-10 |
| \(O^+_8(2)\) | \(-\) | \(-\) | \(15\) | 11.5-12 |
| \(O^+_8(3)\) | \(-\) | \(-\) | \(20\) | 11.5-13 |
| \(O^+_8(4)\) | \(+\) | \(65\) | 11.5-14 | |
| \(O^+_{10}(2)\) | \(+\) | \(45\) | 11.4-6 | |
| \(O^+_{12}(2)\) | \(+\) | \(85\) | 11.4-8 | |
| \(O^+_{12}(3)\) | \(+\) | \(205\) | 11.5-18 | |
| \(O^-_8(2)\) | \(+\) | \(17\) | 11.4-3 | |
| \(O^-_8(3)\) | \(+\) | \(41\) | 11.4-5 | |
| \(O^-_{10}(2)\) | \(+\) | \(33\) | 11.4-7 | |
| \(O^-_{10}(3)\) | \(+\) | \(122\) | 11.5-16 | |
| \(O^-_{12}(2)\) | \(+\) | \(65\) | 11.4-9 | |
| \(O^-_{14}(2)\) | \(+\) | \(129\) | 11.5-17 | |
| \(O_7(3)\) | \(-\) | \(-\) | \(14\) | 11.5-11 |
| \(S_4(4)\) | \(+\) | \(17\) | 11.4-3, 11.4-4 | |
| \(S_6(2)\) | \(-\) | \(-\) | \(9\) | 11.5-20 |
| \(S_6(3)\) | \(+\) | \(14\) | 11.4-3, 11.4-4 | |
| \(S_6(4)\) | \(+\) | \(65\) | 11.4-10 | |
| \(S_8(2)\) | \(-\) | \(-\) | \(17\) | 11.5-21 |
| \(S_8(3)\) | \(+\) | \(41\) | 11.4-12 | |
| \(U_3(3)\) | \(+\) | \(6\) | 11.4-3, 11.4-4 | |
| \(U_3(5)\) | \(+\) | \(10\) | 11.4-3, 11.4-4 | |
| \(U_4(2)\) | \(-\) | \(-\) | \(9\) | 11.5-23 |
| \(U_4(3)\) | \(-\) | \(+\) | \(7\) | 11.5-24 |
| \(U_4(4)\) | \(+\) | \(65\) | 11.4-13 | |
| \(U_5(2)\) | \(+\) | \(11\) | 11.4-3 | |
| \(U_6(2)\) | \(+\) | \(11\) | 11.4-14 | |
| \(U_6(3)\) | \(+\) | \(122\) | 11.5-25 | |
| \(U_8(2)\) | \(+\) | \(129\) | 11.5-26 |
Contrary to [BGK08], Atlas notation is used throughout this note, because the identifiers used for character tables in the GAP Character Table Library follow mainly the Atlas [CCN+85]. For example, we write \(L_d(q)\) for \(PSL(d,q)\), \(S_d(q)\) for \(PSp(d,q)\), \(U_d(q)\) for \(PSU(d,q)\), and \(O^+_{2d}(q)\), \(O^-_{2d}(q)\), \(O_{2d+1}(q)\) for \(PΩ^+(2d,q)\), \(PΩ^-(2d,q)\), \(PΩ(2d+1,q)\), respectively.
Furthermore, in the case of classical groups, the character tables of the (almost) simple groups are considered not the tables of the matrix groups (which are in fact often not available in the GAP Character Table Library). Consequently, also element orders and the description of maximal subgroups refer to the (almost) simple groups not to the matrix groups.
This note contains also several examples that are not needed for the proofs in [BGK08]. Besides several small simple groups \(G\) whose character table is contained in the GAP Character Table Library and for which enough information is available for computing \(σ(G)\), in Section 11.4-3, a few such examples appear in individual sections. In the table of contents, the section headers of the latter kind of examples are marked with an asterisk \((\ast)\).
The examples use the GAP Character Table Library, the GAP Library of Tables of Marks, and the GAP interface [WPN+22] to the Atlas of Group Representations [WWT+], so we first load these three packages in the required versions. The GAP output was adjusted to the versions shown below; in older versions, features necessary for the computations may be missing, and it may happen that with newer versions, the behaviour is different.
gap> CompareVersionNumbers( GAPInfo.Version, "4.5.0" ); true gap> LoadPackage( "ctbllib", "1.2", false ); true gap> LoadPackage( "tomlib", "1.2", false ); true gap> LoadPackage( "atlasrep", "1.5", false ); true
Some of the computations in Section 11.5 require about \(800\) MB of space (on \(32\) bit machines). Therefore we check whether GAP was started with sufficient maximal memory; the command line option for this is -o 800m.
gap> max:= GAPInfo.CommandLineOptions.o;; gap> if not ( ( IsSubset( max, "m" ) and > Int( Filtered( max, IsDigitChar ) ) >= 800 ) or > ( IsSubset( max, "g" ) and > Int( Filtered( max, IsDigitChar ) ) >= 1 ) ) then > Print( "the maximal allowed memory might be too small\n" ); > fi;
Several computations involve calls to the GAP function Random (Reference: Random). In order to make the results of individual examples reproducible, independent of the rest of the computations, we reset the relevant random number generators whenever this is appropriate. For that, we store the initial states in the variable staterandom, and provide a function for resetting the random number generators. (The Random (Reference: Random) calls in the GAP library use the two random number generators GlobalRandomSource (Reference: GlobalRandomSource) and GlobalMersenneTwister (Reference: GlobalMersenneTwister).)
gap> staterandom:= [ State( GlobalRandomSource ), > State( GlobalMersenneTwister ) ];; gap> ResetGlobalRandomNumberGenerators:= function() > Reset( GlobalRandomSource, staterandom[1] ); > Reset( GlobalMersenneTwister, staterandom[2] ); > end;;
Let \(G\) be a finite group, \(S\) the socle of \(G\), and denote by \(G^{\times}\) the set of nonidentity elements in \(G\). For \(s, g \in G^{\times}\), let \(P( g, s ):= |\{ h \in G; S \nsubseteq \langle s^h, g \rangle \}| / |G|\), the proportion of elements in the class \(s^G\) which fail to generate at least \(S\) with \(g\); we set \(P( G, s ):= \max\{ P( g, s ); g \in G^{\times} \}\). We are interested in finding a class \(s^G\) of elements in \(S\) such that \(P( G, s ) < 1/3\) holds.
First consider \(g \in S\), and let \(𝕄(S,s)\) denote the set of those maximal subgroups of \(S\) that contain \(s\). We have
\[ |\{ h \in S; S \nsubseteq \langle s^h, g \rangle \}| = |\{ h \in S; \langle s, h g h^{-1} \rangle ≠ S \}| \\ \leq \sum_{M \in 𝕄(S,s)} |\{ h \in S; h g h^{-1} \in M \}| \]
Since \(h g h^{-1} \in M\) holds if and only if the coset \(M h\) is fixed by \(g\) under the permutation action of \(S\) on the right cosets of \(M\) in \(S\), we get that \(|\{ h \in S; h g h^{-1} \in M \}| = |C_S(g)| \cdot |g^S \cap M| = |M| \cdot 1_M^S(g)\), where \(1_M^S\) is the permutation character of this action, of degree \(|S|/|M|\). Thus
\[ |\{ h \in S; \langle s, h g h^{-1} \rangle ≠ S \}| / |S| \leq \sum_{M \in 𝕄(S,s)} 1_M^S(g) / 1_M^S(1) . \]
We abbreviate the right hand side of this inequality by \(σ( g, s )\), set \(σ( S, s ):= \max\{ σ( g, s ); g \in S^{\times} \}\), and choose a transversal \(T\) of \(S\) in \(G\). Then \(P( g, s ) \leq |T|^{-1} \cdot \sum_{t \in T} σ( g^t, s )\) and thus \(P( G, s ) \leq σ( S, s )\) holds.
If \(S = G\) and if \(𝕄(G,s)\) consists of a single maximal subgroup \(M\) of \(G\) then equality holds, i.e., \(P( g, s ) = σ( g, s ) = 1_M^S(g) / 1_M^S(1)\).
The quantity \(1_M^S(g) / 1_M^S(1) = |g^S \cap M| / |g^S|\) is the proportion of fixed points of \(g\) in the permutation action of \(S\) on the right cosets of its subgroup \(M\). This is called the fixed point ratio of \(g\) w. r. t. \(S/M\), and is denoted as \(μ(g,S/M)\).
For a subgroup \(M\) of \(S\), the number \(n\) of \(S\)-conjugates of \(M\) containing \(s\) is equal to \(|M^S| \cdot |s^S \cap M| / |s^S|\). To see this, consider the set \(\{ (s^h, M^k); h, k \in S, s^h \in M^k \}\), the cardinality of which can be counted either as \(|M^S| \cdot |s^S \cap M|\) or as \(|s^S| \cdot n\). So we get \(n = |M| \cdot 1_M^S(s) / |N_S(M)|\).
If \(S\) is a finite nonabelian simple group then each maximal subgroup in \(S\) is self-normalizing, and we have \(n = 1_M^S(s)\) if \(M\) is maximal. So we can replace the summation over \(𝕄(S,s)\) by one over a set \(𝕄/~(S,s)\) of representatives of conjugacy classes of maximal subgroups of \(S\), and get that
\[ σ( g, s ) = \sum_{M \in 𝕄/~(S,s)} \frac{1_M^S(s) \cdot 1_M^S(g)}{1_M^S(1)}. \]
Furthermore, we have \(|𝕄(S,s)| = \sum_{M \in 𝕄/~(S,s)} 1_M^S(s)\).
In the following, we will often deal with the quantities \(σ(S):= \min\{ σ( S, s ); s \in S^{\times} \}\) and \(𝕊/~(S):= \lceil 1 / σ(S) - 1 \rceil\). These values can be computed easily from the primitive permutation characters of \(S\).
Analogously, we set \(P(S):= \min \{ P( S, s ); s \in S^{\times} \}\) and \(P(S):= \lceil 1 / P(S) - 1 \rceil\). Clearly we have \(P(S) \leq σ(S)\) and \(P(S) \geq 𝕊/~(S)\).
One interpretation of \(P(S)\) is that if this value is at least \(k\) then it follows that for any \(g_1, g_2, \ldots, g_k \in S^{\times}\), there is some \(s \in S\) such that \(S = \langle g_i, s \rangle\), for \(1 \leq i \leq k\). In this case, \(S\) is said to have spread at least \(k\). (Note that the lower bound \(𝕊/~(S)\) for \(P(S)\) can be computed from the list of primitive permutation characters of \(S\).)
Moreover, \(P(S) \geq k\) implies that the element \(s\) can be chosen uniformly from a fixed conjugacy class of \(S\). This is called uniform spread at least \(k\) in [BGK08].
It is proved in [GK00] that all finite simple groups have uniform spread at least \(1\), that is, for any element \(x \in S^{\times}\), there is an element \(y\) in a prescribed class of \(S\) such that \(G = \langle x, y \rangle\) holds. In [BGK08, Corollary 1.3], it is shown that all finite simple groups have uniform spread at least \(2\), and the finite simple groups with (uniform) spread exactly \(2\) are listed.
Concerning the spread, it should be mentioned that the methods used here and in [BGK08] are nonconstructive in the sense that they cannot be used for finding an element \(s\) that generates \(G\) together with each of the \(k\) prescribed elements \(g_1, g_2, \ldots, g_k\).
Now consider \(g \in G \setminus S\). Since \(P( g^k, s ) \geq P( g, s )\) for any positive integer \(k\), we can assume that \(g\) has prime order \(p\), say. We set \(H = \langle S, g \rangle \leq G\), with \([H:S] = p\), choose a transversal \(T\) of \(H\) in \(G\), let \(𝕄^{\prime}(H,s):= 𝕄(H,s) \setminus \{ S \}\), and let \(𝕄/~^{\prime}(H,s)\) denote a set of representatives of \(H\)-conjugacy classes of these groups. As above,
| \(|\{ h \in H; S \nsubseteq \langle s^h, g \rangle \}| / |H|\) | \(=\) | \(|\{ h \in H; \langle s^h, g \rangle ≠ H \}| / |H|\) |
| \(\leq\) | \(\sum_{M \in 𝕄^{\prime}(H,s)} |\{ h \in H; h g h^{-1} \in M \}| / |H|\) | |
| \(=\) | \(\sum_{M \in 𝕄^{\prime}(H,s)} 1_M^H(g) / 1_M^H(1)\) | |
| \(=\) | \(\sum_{M \in 𝕄/~^{\prime}(H,s)} 1_M^H(g) \cdot 1_M^H(s) / 1_M^H(1)\) |
(Note that no summand for \(M = S\) occurs, so each group in \(𝕄/~^{\prime}(H,s)\) is self-normalizing.) We abbreviate the right hand side by \(σ(H,g,s)\), and set \(σ^{\prime}( H, s ) = \max\{ σ(H,g,s); g \in H \setminus S, |g| = [H:S] \}\). Then we get \(P( g, s ) \leq |T|^{-1} \cdot \sum_{t \in T} σ(H^t,g^t,s)\) and thus
\[ P( G, s ) \leq \max\{ P( S, s ), \max\{ σ^{\prime}( H, s ); S \leq H \leq G, [H:S] \textrm{\ prime} \} \} . \]
For convenience, we set \(P^{\prime}(G,s) = \max\{ P(g,s); g \in G \setminus S \}\).
The following criteria will be used when we have to show the existence or nonexistence of \(x_1, x_2, \ldots, x_k\), and \(s \in G\) with the property \(\langle x_i, s \rangle = G\) for \(1 \leq i \leq k\). Note that manipulating lists of integers (representing fixed or moved points) is much more efficient than testing whether certain permutations generate a given group.
Lemma:
Let \(G\) be a finite group, \(s \in G^{\times}\), and \(X = \bigcup_{M \in 𝕄(G,s)} G/M\). For \(x_1, x_2, \ldots, x_k \in G\), the conjugate \(s^{\prime}\) of \(s\) satisfies \(\langle x_i, s^{\prime} \rangle = G\) for \(1 \leq i \leq k\) if and only if \(Fix_{X}(s^{\prime}) \cap \bigcup_{i=1}^k Fix_{X}(x_i) = \emptyset\) holds.
Proof. If \(s^g \in U \leq G\) for some \(g \in G\) then \(Fix_{X}(U) = \emptyset\) if and only if \(U = G\) holds; note that \(Fix_{X}(G) = \emptyset\), and \(Fix_{X}(U) = \emptyset\) implies that \(U \nsubseteq h^{-1} M h\) holds for all \(h \in G\) and \(M \in 𝕄(G,s)\), thus \(U = G\). Applied to \(U = \langle x_i, s^{\prime} \rangle\), we get \(\langle x_i, s^{\prime} \rangle = G\) if and only if \(Fix_{X}(s^{\prime}) \cap Fix_{X}(x_i) = Fix_{X}(U) = \emptyset\).
Corollary 1:
If \(𝕄(G,s) = \{ M \}\) in the situation of the above Lemma then there is a conjugate \(s^{\prime}\) of \(s\) that satisfies \(\langle x_i, s^{\prime} \rangle = G\) for \(1 \leq i \leq k\) if and only if \(\bigcup_{i=1}^k Fix_{X}(x_i) ≠ X\).
Corollary 2:
Let \(G\) be a finite simple group and let \(X\) be a \(G\)-set such that each \(g \in G\) fixes at least one point in \(X\) but that \(Fix_{X}(G) = \emptyset\) holds. If \(x_1, x_2, \ldots x_k\) are elements in \(G\) such that \(\bigcup_{i=1}^k Fix_{X}(x_i) = X\) holds then for each \(s \in G\) there is at least one \(i\) with \(\langle x_i, s \rangle ≠ G\).
After the introduction of general utilities in Section 11.3-1, we distinguish two different tasks. Section 11.3-2 introduces functions that will be used in the following to compute \(σ(g,s)\) with character-theoretic methods. Functions for computing \(P(g,s)\) or an upper bound for this value will be introduced in Section 11.3-3.
The GAP functions shown in this section are collected in the file tst/probgen.g that is distributed with the GAP Character Table Library, see http://www.math.rwth-aachen.de/~Thomas.Breuer/ctbllib.
The functions have been designed for the examples in the later sections, they could be generalized and optimized for other examples. It is not our aim to provide a package for this functionality.
Let list be a dense list and prop be a unary function that returns true or false when applied to the entries of list. PositionsProperty returns the set of positions in list for which true is returned.
gap> if not IsBound( PositionsProperty ) then > PositionsProperty:= function( list, prop ) > return Filtered( [ 1 .. Length( list ) ], i -> prop( list[i] ) ); > end; > fi;
The following two functions implement loops over ordered triples (and quadruples, respectively) in a Cartesian product. A prescribed function prop is subsequently applied to the triples (quadruples), and if the result of this call is true then this triple (quadruple) is returned immediately; if none of the calls to prop yields true then fail is returned.
gap> BindGlobal( "TripleWithProperty", function( threelists, prop ) > local i, j, k, test; > > for i in threelists[1] do > for j in threelists[2] do > for k in threelists[3] do > test:= [ i, j, k ]; > if prop( test ) then > return test; > fi; > od; > od; > od; > > return fail; > end ); gap> BindGlobal( "QuadrupleWithProperty", function( fourlists, prop ) > local i, j, k, l, test; > > for i in fourlists[1] do > for j in fourlists[2] do > for k in fourlists[3] do > for l in fourlists[4] do > test:= [ i, j, k, l ]; > if prop( test ) then > return test; > fi; > od; > od; > od; > od; > > return fail; > end );
Of course one could do better by considering unordered \(n\)-tuples when several of the argument lists are equal, and in practice, backtrack searches would often allow one to prune parts of the search tree in early stages. However, the above loops are not time critical in the examples presented here, so the possible improvements are not worth the effort for our purposes.
The function PrintFormattedArray prints the matrix array in a columnwise formatted way. (The only diference to the GAP library function PrintArray (Reference: PrintArray) is that PrintFormattedArray chooses each column width according to the entries only in this column not w.r.t. the whole matrix.)
gap> BindGlobal( "PrintFormattedArray", function( array ) > local colwidths, n, row; > array:= List( array, row -> List( row, String ) ); > colwidths:= List( TransposedMat( array ), > col -> Maximum( List( col, Length ) ) ); > n:= Length( array[1] ); > for row in List( array, row -> List( [ 1 .. n ], > i -> String( row[i], colwidths[i] ) ) ) do > Print( " ", JoinStringsWithSeparator( row, " " ), "\n" ); > od; > end );
Finally, CleanWorkspace is a utility for reducing the space needed. This is achieved by unbinding those user variables that are not write protected and are not mentioned in the list NeededVariables of variable names that are bound now, and by flushing the caches of tables of marks and character tables.
gap> BindGlobal( "NeededVariables", NamesUserGVars() ); gap> BindGlobal( "CleanWorkspace", function() > local name, record; > > for name in Difference( NamesUserGVars(), NeededVariables ) do > if not IsReadOnlyGlobal( name ) then > UnbindGlobal( name ); > fi; > od; > for record in [ LIBTOMKNOWN, LIBTABLE ] do > for name in RecNames( record.LOADSTATUS ) do > Unbind( record.LOADSTATUS.( name ) ); > Unbind( record.( name ) ); > od; > od; > end );
The function PossiblePermutationCharacters takes two ordinary character tables sub and tbl, computes the possible class fusions from sub to tbl, then induces the trivial character of sub to tbl, w.r.t. these fusions, and returns the set of these class functions. (So if sub and tbl are the character tables of groups \(H\) and \(G\), respectively, where \(H\) is a subgroup of \(G\), then the result contains the permutation character \(1_H^G\).)
Note that the columns of the character tables in the GAP Character Table Library are not explicitly associated with particular conjugacy classes of the corresponding groups, so from the character tables, we can compute only possible class fusions, i.e., maps between the columns of two tables that satisfy certain necessary conditions, see the section about the function PossibleClassFusions in the GAP Reference Manual for details. There is no problem if the permutation character is uniquely determined by the character tables, in all other cases we give ad hoc arguments for resolving the ambiguities.
gap> if not IsBound( PossiblePermutationCharacters ) then > BindGlobal( "PossiblePermutationCharacters", function( sub, tbl ) > local fus, triv; > > fus:= PossibleClassFusions( sub, tbl ); > if fus = fail then > return fail; > fi; > triv:= [ TrivialCharacter( sub ) ]; > > return Set( > List( fus, map -> Induced( sub, tbl, triv, map )[1] ) ); > end ); > fi;
We want to use the GAP libraries of character tables and of tables of marks, and proceed in three steps.
First we extract the primitive permutation characters from the library information if this is available; for that, we write the function PrimitivePermutationCharacters. Then the result can be used as the input for the function ApproxP, which computes the values \(σ( g, s )\). Finally, the functions ProbGenInfoSimple and ProbGenInfoAlmostSimple compute \(𝕊/~( G )\).
For a group \(G\) whose character table \(T\) is contained in the GAP character table library, the complete set of primitive permutation characters can be easily computed if the character tables of all maximal subgroups and their class fusions into \(T\) are known (in this case, we check whether the attribute Maxes (CTblLib: Maxes) of \(T\) is bound) or if the table of marks of \(G\) and the class fusion from \(T\) into this table of marks are known (in this case, we check whether the attribute FusionToTom (CTblLib: FusionToTom) of \(T\) is bound). If the attribute UnderlyingGroup (Reference: UnderlyingGroup for tables of marks) of \(T\) is bound then this group can be used to compute the primitive permutation characters. The latter happens if \(T\) was computed from the group object in GAP; for tables in the GAP character table library, this is not the case by default.
The GAP function PrimitivePermutationCharacters tries to compute the primitive permutation characters of a group using this information; it returns the required list of characters if this can be computed this way, otherwise fail is returned. (For convenience, we use the GAP mechanism of attributes in order to store the permutation characters in the character table object once they have been computed.)
gap> DeclareAttribute( "PrimitivePermutationCharacters", IsCharacterTable ); gap> InstallOtherMethod( PrimitivePermutationCharacters, > [ IsCharacterTable ], > function( tbl ) > local maxes, tom, G; > > if HasMaxes( tbl ) then > maxes:= List( Maxes( tbl ), CharacterTable ); > if ForAll( maxes, s -> GetFusionMap( s, tbl ) <> fail ) then > return List( maxes, subtbl -> TrivialCharacter( subtbl )^tbl ); > fi; > elif HasFusionToTom( tbl ) then > tom:= TableOfMarks( tbl ); > maxes:= MaximalSubgroupsTom( tom ); > return PermCharsTom( tbl, tom ){ maxes[1] }; > elif HasUnderlyingGroup( tbl ) then > G:= UnderlyingGroup( tbl ); > return List( MaximalSubgroupClassReps( G ), > M -> TrivialCharacter( M )^tbl ); > fi; > > return fail; > end );
The function ApproxP takes a list primitives of primitive permutation characters of a group \(G\), say, and the position spos of the class \(s^G\) in the character table of \(G\).
Assume that the elements in primitives have the form \(1_M^G\), for suitable maximal subgroups \(M\) of \(G\), and let \(𝕄/~\) be the set of these groups \(M\). ApproxP returns the class function \(\psi\) of \(G\) that is defined by \(\psi(1) = 0\) and
\[ \psi(g) = \sum_{M \in 𝕄/~} \frac{1_M^G(s) \cdot 1_M^G(g)}{1_M^G(1)} \]
otherwise.
If primitives contains all those primitive permutation characters \(1_M^G\) of \(G\) (with multiplicity according to the number of conjugacy classes of these maximal subgroups) that do not vanish at \(s\), and if all these \(M\) are self-normalizing in \(G\) –this holds for example if \(G\) is a finite simple group– then \(\psi(g) = σ( g, s )\) holds.
gap> BindGlobal( "ApproxP", function( primitives, spos ) > local sum; > > sum:= ShallowCopy( Sum( List( primitives, > pi -> pi[ spos ] * pi / pi[1] ) ) ); > sum[1]:= 0; > > return sum; > end );
Note that for computations with permutation characters, it would make the functions more complicated (and not more efficient) if we would consider only elements \(g\) of prime order, and only one representative of Galois conjugate classes.
The next functions needed in this context compute \(σ(S)\) and \(𝕊/~( S )\), for a simple group \(S\), and \(σ^{\prime}(G,s)\) for an almost simple group \(G\) with socle \(S\), respectively.
ProbGenInfoSimple takes the character table tbl of \(S\) as its argument. If the full list of primitive permutation characters of \(S\) cannot be computed with PrimitivePermutationCharacters then the function returns fail. Otherwise ProbGenInfoSimple returns a list containing the identifier of the table, the value \(σ(S)\), the integer \(𝕊/~( S )\), a list of Atlas names of representatives of Galois families of those classes of elements \(s\) for which \(σ(S) = σ( S, s )\) holds, and the list of the corresponding cardinalities \(|𝕄(S,s)|\).
gap> BindGlobal( "ProbGenInfoSimple", function( tbl ) > local prim, max, min, bound, s; > prim:= PrimitivePermutationCharacters( tbl ); > if prim = fail then > return fail; > fi; > max:= List( [ 1 .. NrConjugacyClasses( tbl ) ], > i -> Maximum( ApproxP( prim, i ) ) ); > min:= Minimum( max ); > bound:= Inverse( min ); > if IsInt( bound ) then > bound:= bound - 1; > else > bound:= Int( bound ); > fi; > s:= PositionsProperty( max, x -> x = min ); > s:= List( Set( s, i -> ClassOrbit( tbl, i ) ), i -> i[1] ); > return [ Identifier( tbl ), > min, > bound, > AtlasClassNames( tbl ){ s }, > Sum( List( prim, pi -> pi{ s } ) ) ]; > end );
ProbGenInfoAlmostSimple takes the character tables tblS and tblG of \(S\) and \(G\), and a list sposS of class positions (w.r.t. tblS) as its arguments. It is assumed that \(S\) is simple and has prime index in \(G\). If PrimitivePermutationCharacters can compute the full list of primitive permutation characters of \(G\) then the function returns a list containing the identifier of tblG, the maximum \(m\) of \(σ^{\prime}( G, s )\), for \(s\) in the classes described by sposS, a list of Atlas names (in \(G\)) of the classes of elements \(s\) for which this maximum is attained, and the list of the corresponding cardinalities \(|𝕄^{\prime}(G,s)|\). When PrimitivePermutationCharacters returns fail, also ProbGenInfoAlmostSimple returns fail.
gap> BindGlobal( "ProbGenInfoAlmostSimple", function( tblS, tblG, sposS ) > local p, fus, inv, prim, sposG, outer, approx, l, max, min, > s, cards, i, names; > > p:= Size( tblG ) / Size( tblS ); > if not IsPrimeInt( p ) > or Length( ClassPositionsOfNormalSubgroups( tblG ) ) <> 3 then > return fail; > fi; > fus:= GetFusionMap( tblS, tblG ); > if fus = fail then > return fail; > fi; > inv:= InverseMap( fus ); > prim:= PrimitivePermutationCharacters( tblG ); > if prim = fail then > return fail; > fi; > sposG:= Set( fus{ sposS } ); > outer:= Difference( PositionsProperty( > OrdersClassRepresentatives( tblG ), IsPrimeInt ), fus ); > approx:= List( sposG, i -> ApproxP( prim, i ){ outer } ); > if IsEmpty( outer ) then > max:= List( approx, x -> 0 ); > else > max:= List( approx, Maximum ); > fi; > min:= Minimum( max); > s:= sposG{ PositionsProperty( max, x -> x = min ) }; > cards:= List( prim, pi -> pi{ s } ); > for i in [ 1 .. Length( prim ) ] do > # Omit the character that is induced from the simple group. > if ForAll( prim[i], x -> x = 0 or x = prim[i][1] ) then > cards[i]:= 0; > fi; > od; > names:= AtlasClassNames( tblG ){ s }; > Perform( names, ConvertToStringRep ); > > return [ Identifier( tblG ), > min, > names, > Sum( cards ) ]; > end );
The next function computes \(σ(G,s)\) from the character table tbl of a simple or almost simple group \(G\), the name sname of the class of \(s\) in this table, the list maxes of the character tables of all subgroups \(M\) with \(M \in 𝕄(G,s)\), and the list numpermchars of the numbers of possible permutation characters induced from maxes. If the string "outer" is given as an optional argument then \(G\) is assumed to be an automorphic extension of a simple group \(S\), with \([G:S]\) a prime, and \(σ^{\prime}(G,s)\) is returned. In both situations, the result is fail if the numbers of possible permutation characters induced from maxes do not coincide with the numbers prescribed in numpermchars.
gap> BindGlobal( "SigmaFromMaxes", function( arg ) > local t, sname, maxes, numpermchars, prim, spos, outer; > > t:= arg[1]; > sname:= arg[2]; > maxes:= arg[3]; > numpermchars:= arg[4]; > prim:= List( maxes, s -> PossiblePermutationCharacters( s, t ) ); > spos:= Position( AtlasClassNames( t ), sname ); > if ForAny( [ 1 .. Length( maxes ) ], > i -> Length( prim[i] ) <> numpermchars[i] ) then > return fail; > elif Length( arg ) = 5 and arg[5] = "outer" then > outer:= Difference( > PositionsProperty( OrdersClassRepresentatives( t ), IsPrimeInt ), > ClassPositionsOfDerivedSubgroup( t ) ); > return Maximum( ApproxP( Concatenation( prim ), spos ){ outer } ); > else > return Maximum( ApproxP( Concatenation( prim ), spos ) ); > fi; > end );
The following function allows us to extract information about \(𝕄(G,s)\) from the character table tbl of \(G\) and a list snames of class positions of \(s\). If Maxes( tbl ) is stored then the names of the character tables of the subgroups in \(𝕄(G,s)\) and the number of conjugates are printed, otherwise fail is printed.
gap> BindGlobal( "DisplayProbGenMaxesInfo", function( tbl, snames ) > local mx, prim, i, spos, nonz, indent, j; > > if not HasMaxes( tbl ) then > Print( Identifier( tbl ), ": fail\n" ); > return; > fi; > > mx:= List( Maxes( tbl ), CharacterTable ); > prim:= List( mx, s -> TrivialCharacter( s )^tbl ); > Assert( 1, SortedList( prim ) = > SortedList( PrimitivePermutationCharacters( tbl ) ) ); > for i in [ 1 .. Length( prim ) ] do > # Deal with the case that the subgroup is normal. > if ForAll( prim[i], x -> x = 0 or x = prim[i][1] ) then > prim[i]:= prim[i] / prim[i][1]; > fi; > od; > > spos:= List( snames, > nam -> Position( AtlasClassNames( tbl ), nam ) ); > nonz:= List( spos, x -> PositionsProperty( prim, pi -> pi[x] <> 0 ) ); > for i in [ 1 .. Length( spos ) ] do > Print( Identifier( tbl ), ", ", snames[i], ": " ); > indent:= ListWithIdenticalEntries( > Length( Identifier( tbl ) ) + Length( snames[i] ) + 4, ' ' ); > if not IsEmpty( nonz[i] ) then > Print( Identifier( mx[ nonz[i][1] ] ), " (", > prim[ nonz[i][1] ][ spos[i] ], ")\n" ); > for j in [ 2 .. Length( nonz[i] ) ] do > Print( indent, Identifier( mx[ nonz[i][j] ] ), " (", > prim[ nonz[i][j] ][ spos[i] ], ")\n" ); > od; > else > Print( "\n" ); > fi; > od; > end );
Here, the task is to compute \(P(g,s)\) or \(P(G,s)\) using explicit computations with \(G\), where the character-theoretic bounds are not sufficient.
We start with small utilities that make the examples shorter.
For a finite solvable group G, the function PcConjugacyClassReps returns a list of representatives of the conjugacy classes of G, which are computed using a polycyclic presentation for G.
gap> BindGlobal( "PcConjugacyClassReps", function( G ) > local iso; > > iso:= IsomorphismPcGroup( G ); > return List( ConjugacyClasses( Image( iso ) ), > c -> PreImagesRepresentative( iso, Representative( c ) ) ); > end );
For a finite group G, a list primes of prime integers, and a normal subgroup N of G, the function ClassesOfPrimeOrder returns a list of those conjugacy classes of G that are not contained in N and whose elements' orders occur in primes.
For each prime \(p\) in primes, first class representatives of order \(p\) in a Sylow \(p\) subgroup of G are computed, then the representatives in N are discarded, and then representatives w. r. t. conjugacy in G are computed.
(Note that this approach may be inappropriate for example if a large elementary abelian Sylow \(p\) subgroup occurs, and if the conjugacy tests in G are expensive, see Section 11.5-14.)
gap> BindGlobal( "ClassesOfPrimeOrder", function( G, primes, N ) > local ccl, p, syl, Greps, reps, r, cr; > > ccl:= []; > for p in primes do > syl:= SylowSubgroup( G, p ); > Greps:= []; > reps:= Filtered( PcConjugacyClassReps( syl ), > r -> Order( r ) = p and not r in N ); > for r in reps do > cr:= ConjugacyClass( G, r ); > if ForAll( Greps, c -> c <> cr ) then > Add( Greps, cr ); > fi; > od; > Append( ccl, Greps ); > od; > > return ccl; > end );
The function IsGeneratorsOfTransPermGroup takes a transitive permutation group G and a list list of elements in G, and returns true if the elements in list generate G, and false otherwise. The main point is that the return value true requires the group generated by list to be transitive, and the check for transitivity is much cheaper than the test whether this group is equal to G.
gap> if not IsBound( IsGeneratorsOfTransPermGroup) then > BindGlobal( "IsGeneratorsOfTransPermGroup", function( G, list ) > local S; > > if not IsTransitive( G ) then > Error( "<G> must be transitive on its moved points" ); > fi; > S:= SubgroupNC( G, list ); > > return IsTransitive( S, MovedPoints( G ) ) and > Size( S ) = Size( G ); > end ); > fi;
RatioOfNongenerationTransPermGroup takes a transitive permutation group G and two elements g and s of G, and returns the proportion \(P(g,s)\). (The function tests the (non)generation only for representatives of \(C_G(g)\)-\(C_G(s)\)-double cosets. Note that for \(c_1 \in C_G(g)\), \(c_2 \in C_G(s)\), and a representative \(r \in G\), we have \(\langle g^{c_1 r c_2}, s \rangle = \langle g^r, s \rangle^{c_2}\).)
gap> BindGlobal( "RatioOfNongenerationTransPermGroup", function( G, g, s ) > local nongen, pair; > > if not IsTransitive( G ) then > Error( "<G> must be transitive on its moved points" ); > fi; > nongen:= 0; > for pair in DoubleCosetRepsAndSizes( G, Centralizer( G, g ), > Centralizer( G, s ) ) do > if not IsGeneratorsOfTransPermGroup( G, [ s, g^pair[1] ] ) then > nongen:= nongen + pair[2]; > fi; > od; > > return nongen / Size( G ); > end );
Let \(G\) be a group, and let groups be a list \([ G_1, G_2, \ldots, G_n ]\) of permutation groups such that \(G_i\) describes the action of \(G\) on a set \(\Omega_i\), say. Moreover, we require that for \(1 \leq i, j \leq n\), mapping the GeneratorsOfGroup list of \(G_i\) to that of \(G_j\) defines an isomorphism. DiagonalProductOfPermGroups takes groups as its argument, and returns the action of \(G\) on the disjoint union of \(\Omega_1, \Omega_2, \ldots, \Omega_n\).
gap> BindGlobal( "DiagonalProductOfPermGroups", function( groups ) > local prodgens, deg, i, gens, D, pi; > > prodgens:= GeneratorsOfGroup( groups[1] ); > deg:= NrMovedPoints( prodgens ); > for i in [ 2 .. Length( groups ) ] do > gens:= GeneratorsOfGroup( groups[i] ); > D:= MovedPoints( gens ); > pi:= MappingPermListList( D, [ deg+1 .. deg+Length( D ) ] ); > deg:= deg + Length( D ); > prodgens:= List( [ 1 .. Length( prodgens ) ], > i -> prodgens[i] * gens[i]^pi ); > od; > > return Group( prodgens ); > end );
The following two functions are used to reduce checks of generation to class representatives of maximal order. Note that if \(\langle s, g \rangle\) is a proper subgroup of \(G\) then also \(\langle s^k, g \rangle\) is a proper subgroup of \(G\), so we need not check powers \(s^k\) different from \(s\) in this situation.
For an ordinary character table tbl, the function RepresentativesMaximallyCyclicSubgroups returns a list of class positions, containing one class of generators for each class of maximally cyclic subgroups.
gap> BindGlobal( "RepresentativesMaximallyCyclicSubgroups", function( tbl ) > local n, result, orders, p, pmap, i, j; > > # Initialize. > n:= NrConjugacyClasses( tbl ); > result:= BlistList( [ 1 .. n ], [ 1 .. n ] ); > > # Omit powers of smaller order. > orders:= OrdersClassRepresentatives( tbl ); > for p in PrimeDivisors( Size( tbl ) ) do > pmap:= PowerMap( tbl, p ); > for i in [ 1 .. n ] do > if orders[ pmap[i] ] < orders[i] then > result[ pmap[i] ]:= false; > fi; > od; > od; > > # Omit Galois conjugates. > for i in [ 1 .. n ] do > if result[i] then > for j in ClassOrbit( tbl, i ) do > if i <> j then > result[j]:= false; > fi; > od; > fi; > od; > > # Return the result. > return ListBlist( [ 1 .. n ], result ); > end );
Let G be a finite group, tbl be the ordinary character table of G, and cols be a list of class positions in tbl, for example the list returned by RepresentativesMaximallyCyclicSubgroups. The function ClassesPerhapsCorrespondingToTableColumns returns the sublist of those conjugacy classes of G for which the corresponding column in tbl can be contained in cols, according to element order and class size.
gap> BindGlobal( "ClassesPerhapsCorrespondingToTableColumns", > function( G, tbl, cols ) > local orders, classes, invariants; > > orders:= OrdersClassRepresentatives( tbl ); > classes:= SizesConjugacyClasses( tbl ); > invariants:= List( cols, i -> [ orders[i], classes[i] ] ); > > return Filtered( ConjugacyClasses( G ), > c -> [ Order( Representative( c ) ), Size(c) ] in invariants ); > end );
The next function computes, for a finite group \(G\) and subgroups \(M_1, M_2, \ldots, M_n\) of \(G\), an upper bound for \(\max \{ \sum_{i=1}^n μ(g,G/M_i); g \in G \setminus Z(G) \}\). So if the \(M_i\) are the groups in \(𝕄(G,s)\), for some \(s \in G^{\times}\), then we get an upper bound for \(σ(G,s)\).
The idea is that for \(M \leq G\) and \(g \in G\) of order \(p\), we have
\[ μ(g,G/M) = |g^G \cap M| / |g^G| \leq \sum_{h \in C} |h^M| / |g^G| = \sum_{h \in C} |h^M| \cdot |C_G(g)| / |G| , \]
where \(C\) is a set of class representatives \(h \in M\) of all those classes that satisfy \(|h| = p\) and \(|C_G(h)| = |C_G(g)|\), and in the case that \(G\) is a permutation group additionally that \(h\) and \(g\) move the same number of points. (Note that it is enough to consider elements of prime order.)
For computing the maximum of the rightmost term in this inequality, for \(g \in G \setminus Z(G)\), we need not determine the \(G\)-conjugacy of class representatives in \(M\). Of course we pay the price that the result may be larger than the leftmost term. However, if the maximal sum is in fact taken only over a single class representative, we are sure that equality holds. Thus we return a list of length two, containing the maximum of the right hand side of the above inequality and a Boolean value indicating whether this is equal to \(\max \{ μ(g,G/M); g \in G \setminus Z(G) \}\) or just an upper bound.
The arguments for UpperBoundFixedPointRatios are the group G, a list maxesclasses such that the \(i\)-th entry is a list of conjugacy classes of \(M_i\), which covers all classes of prime element order in \(M_i\), and either true or false, where true means that the exact value of \(σ(G,s)\) is computed, not just an upper bound; this can be much more expensive because of the conjugacy tests in \(G\) that may be necessary. (We try to reduce the number of conjugacy tests in this case, the second half of the code is not completely straightforward. The special treatment of conjugacy checks for elements with the same sets of fixed points is essential in the computation of \(σ^{\prime}(G,s)\) for \(G = PGL(6,4)\); the critical input line is ApproxPForOuterClassesInGL( 6, 4 ), see Section 11.5-7. Currently the standard GAP conjugacy test for an element of order three and its inverse in \(G \setminus G^{\prime}\) requires hours of CPU time, whereas the check for existence of a conjugating element in the stabilizer of the common set of fixed points of the two elements is almost free of charge.)
UpperBoundFixedPointRatios can be used to compute \(σ^{\prime}(G,s)\) in the case that \(G\) is an automorphic extension of a simple group \(S\), with \([G:S] = p\) a prime; if \(𝕄^{\prime}(G,s) = \{ M_1, M_2, \ldots, M_n \}\) then the \(i\)-th entry of maxesclasses must contain only the classes of element order \(p\) in \(M_i \setminus (M_i \cap S)\).
gap> BindGlobal( "UpperBoundFixedPointRatios", > function( G, maxesclasses, truetest ) > local myIsConjugate, invs, info, c, r, o, inv, pos, sums, max, maxpos, > maxlen, reps, split, i, found, j; > > myIsConjugate:= function( G, x, y ) > local movx, movy; > > movx:= MovedPoints( x ); > movy:= MovedPoints( y ); > if movx = movy then > G:= Stabilizer( G, movx, OnSets ); > fi; > return IsConjugate( G, x, y ); > end; > > invs:= []; > info:= []; > > # First distribute the classes according to invariants. > for c in Concatenation( maxesclasses ) do > r:= Representative( c ); > o:= Order( r ); > # Take only prime order representatives. > if IsPrimeInt( o ) then > inv:= [ o, Size( Centralizer( G, r ) ) ]; > # Omit classes that are central in `G'. > if inv[2] <> Size( G ) then > if IsPerm( r ) then > Add( inv, NrMovedPoints( r ) ); > fi; > pos:= First( [ 1 .. Length( invs ) ], i -> inv = invs[i] ); > if pos = fail then > # This class is not `G'-conjugate to any of the previous ones. > Add( invs, inv ); > Add( info, [ [ r, Size( c ) * inv[2] ] ] ); > else > # This class may be conjugate to an earlier one. > Add( info[ pos ], [ r, Size( c ) * inv[2] ] ); > fi; > fi; > fi; > od; > > if info = [] then > return [ 0, true ]; > fi; > > repeat > # Compute the contributions of the classes with the same invariants. > sums:= List( info, x -> Sum( List( x, y -> y[2] ) ) ); > max:= Maximum( sums ); > maxpos:= Filtered( [ 1 .. Length( info ) ], i -> sums[i] = max ); > maxlen:= List( maxpos, i -> Length( info[i] ) ); > > # Split the sets with the same invariants if necessary > # and if we want to compute the exact value. > if truetest and not 1 in maxlen then > # Make one conjugacy test. > pos:= Position( maxlen, Minimum( maxlen ) ); > reps:= info[ maxpos[ pos ] ]; > if myIsConjugate( G, reps[1][1], reps[2][1] ) then > # Fuse the two classes. > reps[1][2]:= reps[1][2] + reps[2][2]; > reps[2]:= reps[ Length( reps ) ]; > Unbind( reps[ Length( reps ) ] ); > else > # Split the list. This may require additional conjugacy tests. > Unbind( info[ maxpos[ pos ] ] ); > split:= [ reps[1], reps[2] ]; > for i in [ 3 .. Length( reps ) ] do > found:= false; > for j in split do > if myIsConjugate( G, reps[i][1], j[1] ) then > j[2]:= reps[i][2] + j[2]; > found:= true; > break; > fi; > od; > if not found then > Add( split, reps[i] ); > fi; > od; > > info:= Compacted( Concatenation( info, > List( split, x -> [ x ] ) ) ); > fi; > fi; > until 1 in maxlen or not truetest; > > return [ max / Size( G ), 1 in maxlen ]; > end );
Suppose that \(C_1, C_2, C_3\) are conjugacy classes in \(G\), and that we have to prove, for each \((x_1, x_2, x_3) \in C_1 \times C_2 \times C_3\), the existence of an element \(s\) in a prescribed class \(C\) of \(G\) such that \(\langle x_1, s \rangle = \langle x_2, s \rangle = \langle x_2, s \rangle = G\) holds.
We have to check only representatives under the conjugation action of \(G\) on \(C_1 \times C_2 \times C_3\). For each representative, we try a prescribed number of random elements in \(C\). If this is successful then we are done. The following two functions implement this idea.
For a group \(G\) and a list \([ g_1, g_2, \ldots, g_n ]\) of elements in \(G\), OrbitRepresentativesProductOfClasses returns a list \(R(G, g_1, g_2, \ldots, g_n)\) of representatives of \(G\)-orbits on the Cartesian product \(g_1^G \times g_2^G \times \cdots \times g_n^G\).
The idea behind this function is to choose \(R(G, g_1) = \{ ( g_1 ) \}\) in the case \(n = 1\), and, for \(n > 1\),
\[ R(G, g_1, g_2, \ldots, g_n) = \{ (h_1, h_2, \ldots, h_n) \mid (h_1, h_2, \ldots, h_{n-1}) \in R(G, g_1, g_2, \ldots, g_{n-1}), \\ h_n = g_n^d, \textrm{\ for\ } d \in D \} , \]
where \(D\) is a set of representatives of double cosets \(C_G(g_n) \setminus G / \cap_{i=1}^{n-1} C_G(h_i)\).
gap> BindGlobal( "OrbitRepresentativesProductOfClasses", > function( G, classreps ) > local cents, n, orbreps; > > cents:= List( classreps, x -> Centralizer( G, x ) ); > n:= Length( classreps ); > > orbreps:= function( reps, intersect, pos ) > if pos > n then > return [ reps ]; > fi; > return Concatenation( List( > DoubleCosetRepsAndSizes( G, cents[ pos ], intersect ), > r -> orbreps( Concatenation( reps, [ classreps[ pos ]^r[1] ] ), > Intersection( intersect, cents[ pos ]^r[1] ), pos+1 ) ) ); > end; > > return orbreps( [ classreps[1] ], cents[1], 2 ); > end );
The function RandomCheckUniformSpread takes a transitive permutation group \(G\), a list of class representatives \(g_i \in G\), an element \(s \in G\), and a positive integer \(N\). The return value is true if for each representative of \(G\)-orbits on the product of the classes \(g_i^G\), a good conjugate of \(s\) is found in at most \(N\) random tests.
gap> BindGlobal( "RandomCheckUniformSpread", function( G, classreps, s, try ) > local elms, found, i, conj; > > if not IsTransitive( G, MovedPoints( G ) ) then > Error( "<G> must be transitive on its moved points" ); > fi; > > # Compute orbit representatives of G on the direct product, > # and try to find a good conjugate of s for each representative. > for elms in OrbitRepresentativesProductOfClasses( G, classreps ) do > found:= false; > for i in [ 1 .. try ] do > conj:= s^Random( G ); > if ForAll( elms, > x -> IsGeneratorsOfTransPermGroup( G, [ x, conj ] ) ) then > found:= true; > break; > fi; > od; > if not found then > return elms; > fi; > od; > > return true; > end );
Of course this approach is not suitable for disproving the existence of \(s\), but it is much cheaper than an exhaustive search in the class \(C\). (Typically, \(|C|\) is large whereas the \(|C_i|\) are small.)
The following function can be used to verify that a given \(n\)-tuple \((x_1, x_2, \ldots, x_n)\) of elements in a group \(G\) has the property that for all elements \(g \in G\), at least one \(x_i\) satisfies \(\langle x_i, g \rangle\). The arguments are a transitive permutation group \(G\), a list of class representatives in \(G\), and the \(n\)-tuple in question. The return value is a conjugate \(g\) of the given representatives that has the property if such an element exists, and fail otherwise.
gap> BindGlobal( "CommonGeneratorWithGivenElements", > function( G, classreps, tuple ) > local inter, rep, repcen, pair; > > if not IsTransitive( G, MovedPoints( G ) ) then > Error( "<G> must be transitive on its moved points" ); > fi; > > inter:= Intersection( List( tuple, x -> Centralizer( G, x ) ) ); > for rep in classreps do > repcen:= Centralizer( G, rep ); > for pair in DoubleCosetRepsAndSizes( G, repcen, inter ) do > if ForAll( tuple, > x -> IsGeneratorsOfTransPermGroup( G, [ x, rep^pair[1] ] ) ) then > return rep; > fi; > od; > od; > > return fail; > end );
In this section, we apply the functions introduced in Section 11.3-2 to the character tables of simple groups that are available in the GAP Character Table Library.
Our first examples are the sporadic simple groups, in Section 11.4-1, then their automorphism groups are considered in Section 11.4-2.
Then we consider those other simple groups for which GAP provides enough information for automatically computing an upper bound on \(σ(G,s)\) –see Section 11.4-3– and their automorphic extensions –see Section 11.4-4.
After that, individual groups are considered.
The GAP Character Table Library contains the tables of maximal subgroups of all sporadic simple groups except \(B\) and \(M\), so all primitive permutation characters can be computed via the function PrimitivePermutationCharacters for \(24\) of the \(26\) sporadic simple groups.
gap> sporinfo:= [];; gap> spornames:= AllCharacterTableNames( IsSporadicSimple, true, > IsDuplicateTable, false );; gap> for tbl in List( spornames, CharacterTable ) do > info:= ProbGenInfoSimple( tbl ); > if info <> fail then > Add( sporinfo, info ); > fi; > od;
We show the result as a formatted table.
gap> PrintFormattedArray( sporinfo ); Co1 421/1545600 3671 [ "35A" ] [ 4 ] Co2 1/270 269 [ "23A" ] [ 1 ] Co3 64/6325 98 [ "21A" ] [ 4 ] F3+ 1/269631216855 269631216854 [ "29A" ] [ 1 ] Fi22 43/585 13 [ "16A" ] [ 7 ] Fi23 2651/2416635 911 [ "23A" ] [ 2 ] HN 4/34375 8593 [ "19A" ] [ 1 ] HS 64/1155 18 [ "15A" ] [ 2 ] He 3/595 198 [ "14C" ] [ 3 ] J1 1/77 76 [ "19A" ] [ 1 ] J2 5/28 5 [ "10C" ] [ 3 ] J3 2/153 76 [ "19A" ] [ 2 ] J4 1/1647124116 1647124115 [ "29A" ] [ 1 ] Ly 1/35049375 35049374 [ "37A" ] [ 1 ] M11 1/3 2 [ "11A" ] [ 1 ] M12 1/3 2 [ "10A" ] [ 3 ] M22 1/21 20 [ "11A" ] [ 1 ] M23 1/8064 8063 [ "23A" ] [ 1 ] M24 108/1265 11 [ "21A" ] [ 2 ] McL 317/22275 70 [ "15A", "30A" ] [ 3, 3 ] ON 10/30723 3072 [ "31A" ] [ 2 ] Ru 1/2880 2879 [ "29A" ] [ 1 ] Suz 141/5720 40 [ "14A" ] [ 3 ] Th 2/267995 133997 [ "27A", "27B" ] [ 2, 2 ]
We see that in all these cases, \(σ(G) < 1/2\) and thus \(P( G ) \geq 2\), and all sporadic simple groups \(G\) except \(G = M_{11}\) and \(G = M_{12}\) satisfy \(σ(G) < 1/3\). See 11.5-9 and 11.5-10 for a proof that also these two groups have uniform spread at least three.
The structures and multiplicities of the maximal subgroups containing \(s\) are as follows.
gap> for entry in sporinfo do > DisplayProbGenMaxesInfo( CharacterTable( entry[1] ), entry[4] ); > od; Co1, 35A: (A5xJ2):2 (1) (A6xU3(3)):2 (2) (A7xL2(7)):2 (1) Co2, 23A: M23 (1) Co3, 21A: U3(5).3.2 (2) L3(4).D12 (1) s3xpsl(2,8).3 (1) F3+, 29A: 29:14 (1) Fi22, 16A: 2^10:m22 (1) (2x2^(1+8)):U4(2):2 (1) 2F4(2)' (4) 2^(5+8):(S3xA6) (1) Fi23, 23A: 2..11.m23 (1) L2(23) (1) HN, 19A: U3(8).3_1 (1) HS, 15A: A8.2 (1) 5:4xa5 (1) He, 14C: 2^1+6.psl(3,2) (1) 7^2:2psl(2,7) (1) 7^(1+2):(S3x3) (1) J1, 19A: 19:6 (1) J2, 10C: 2^1+4b:a5 (1) a5xd10 (1) 5^2:D12 (1) J3, 19A: L2(19) (1) J3M3 (1) J4, 29A: frob (1) Ly, 37A: 37:18 (1) M11, 11A: L2(11) (1) M12, 10A: A6.2^2 (1) M12M4 (1) 2xS5 (1) M22, 11A: L2(11) (1) M23, 23A: 23:11 (1) M24, 21A: L3(4).3.2_2 (1) 2^6:(psl(3,2)xs3) (1) McL, 15A: 3^(1+4):2S5 (1) 2.A8 (1) 5^(1+2):3:8 (1) McL, 30A: 3^(1+4):2S5 (1) 2.A8 (1) 5^(1+2):3:8 (1) ON, 31A: L2(31) (1) ONM8 (1) Ru, 29A: L2(29) (1) Suz, 14A: J2.2 (2) (a4xpsl(3,4)):2 (1) Th, 27A: ThN3B (1) ThM7 (1) Th, 27B: ThN3B (1) ThM7 (1)
For the remaining two sporadic simple groups, \(B\) and \(M\), we choose suitable elements \(s\). If \(G = B\) and \(s \in G\) is of order \(47\) then, by [Wil99], \(𝕄(G,s) = \{ 47:23 \}\).
gap> SigmaFromMaxes( CharacterTable( "B" ), "47A", > [ CharacterTable( "47:23" ) ], [ 1 ] ); 1/174702778623598780219392000000
If \(G = M\) and \(s \in G\) is of order \(59\) then, by [HW04], \(𝕄(G,s) = \{ L_2(59) \}\). In this case, the permutation character is not uniquely determined by the character tables, but all possibilities lead to the same value for \(σ(G)\).
gap> t:= CharacterTable( "M" );; gap> s:= CharacterTable( "L2(59)" );; gap> pi:= PossiblePermutationCharacters( s, t );; gap> Length( pi ); 5 gap> spos:= Position( OrdersClassRepresentatives( t ), 59 ); 152 gap> Set( pi, x -> Maximum( ApproxP( [ x ], spos ) ) ); [ 1/3385007637938037777290625 ]
Essentially the same approach is taken in [GM01]. However, there \(s\) is restricted to classes of prime order. Thus the results in the above table are better for \(J_2\), \(HS\), \(M_{24}\), \(McL\), \(He\), \(Suz\), \(Co_3\), \(Fi_{22}\), \(Ly\), \(Th\), \(Co_1\), and \(J_4\). Besides that, the value \(10\,999\) claimed in [GM01] for \(𝕊/~( HN )\) is not correct.
Next we consider the automorphism groups of the sporadic simple groups. There are exactly \(12\) cases where nontrivial outer automorphisms exist, and then the simple group \(S\) has index \(2\) in its automorphism group \(G\).
gap> sporautnames:= AllCharacterTableNames( IsSporadicSimple, true, > IsDuplicateTable, false, > OfThose, AutomorphismGroup );; gap> sporautnames:= Difference( sporautnames, spornames ); [ "F3+.2", "Fi22.2", "HN.2", "HS.2", "He.2", "J2.2", "J3.2", "M12.2", "M22.2", "McL.2", "ON.2", "Suz.2" ]
First we compute the values \(σ^{\prime}(G,s)\), for the same \(s \in S\) that were chosen for the simple group \(S\) in Section 11.4-1.
For six of the groups \(G\) in question, the character tables of all maximal subgroups are available in the GAP Character Table Library. In these cases, the values \(σ^{\prime}( G, s )\) can be computed using ProbGenInfoAlmostSimple.
(The above statement can meanwhile be replaced by the statement that the character tables of all maximal subgroups are available for all twelve groups. We show the table results for all these groups but keep the individual computations from the original computations.)
gap> sporautinfo:= [];; gap> fails:= [];; gap> for name in sporautnames do > tbl:= CharacterTable( name{ [ 1 .. Position( name, '.' ) - 1 ] } ); > tblG:= CharacterTable( name ); > info:= ProbGenInfoSimple( tbl ); > info:= ProbGenInfoAlmostSimple( tbl, tblG, > List( info[4], x -> Position( AtlasClassNames( tbl ), x ) ) ); > if info = fail then > Add( fails, name ); > else > Add( sporautinfo, info ); > fi; > od; gap> PrintFormattedArray( sporautinfo ); F3+.2 0 [ "29AB" ] [ 1 ] Fi22.2 251/3861 [ "16AB" ] [ 7 ] HN.2 1/6875 [ "19AB" ] [ 1 ] HS.2 36/275 [ "15A" ] [ 2 ] He.2 37/9520 [ "14CD" ] [ 3 ] J2.2 1/15 [ "10CD" ] [ 3 ] J3.2 1/1080 [ "19AB" ] [ 1 ] M12.2 4/99 [ "10A" ] [ 1 ] M22.2 1/21 [ "11AB" ] [ 1 ] McL.2 1/63 [ "15AB", "30AB" ] [ 3, 3 ] ON.2 1/84672 [ "31AB" ] [ 1 ] Suz.2 661/46332 [ "14A" ] [ 3 ]
Note that for \(S = McL\), the bound \(σ^{\prime}(G,s)\) for \(G = S.2\) (in the second column) is worse than the bound for the simple group \(S\).
The structures and multiplicities of the maximal subgroups containing \(s\) are as follows.
gap> for entry in sporautinfo do > DisplayProbGenMaxesInfo( CharacterTable( entry[1] ), entry[3] ); > od; F3+.2, 29AB: F3+ (1) frob (1) Fi22.2, 16AB: Fi22 (1) Fi22.2M4 (1) (2x2^(1+8)):(U4(2):2x2) (1) 2F4(2)'.2 (4) 2^(5+8):(S3xS6) (1) HN.2, 19AB: HN (1) U3(8).6 (1) HS.2, 15A: HS (1) S8x2 (1) 5:4xS5 (1) He.2, 14CD: He (1) 2^(1+6)_+.L3(2).2 (1) 7^2:2.L2(7).2 (1) 7^(1+2):(S3x6) (1) J2.2, 10CD: J2 (1) 2^(1+4).S5 (1) (A5xD10).2 (1) 5^2:(4xS3) (1) J3.2, 19AB: J3 (1) 19:18 (1) M12.2, 10A: M12 (1) (2^2xA5):2 (1) M22.2, 11AB: M22 (1) L2(11).2 (1) McL.2, 15AB: McL (1) 3^(1+4):4S5 (1) Isoclinic(2.A8.2) (1) 5^(1+2):(24:2) (1) McL.2, 30AB: McL (1) 3^(1+4):4S5 (1) Isoclinic(2.A8.2) (1) 5^(1+2):(24:2) (1) ON.2, 31AB: ON (1) 31:30 (1) Suz.2, 14A: Suz (1) J2.2x2 (2) (A4xL3(4):2_3):2 (1)
Note that the maximal subgroups \(L_2(19)\) of \(J_3\) do not extend to \(J_3.2\) and that a class of maximal subgroups of the type \(19:18\) appears in \(J_3.2\) whose intersection with \(J_3\) is not maximal in \(J_3\). Similarly, the maximal subgroups \(A_6.2^2\) of \(M_{12}\) do not extend to \(M_{12}.2\).
For the other six groups, we use individual computations.
In the case \(S = Fi_{24}^{\prime}\), the unique maximal subgroup \(29:14\) that contains an element \(s\) of order \(29\) extends to a group of the type \(29:28\) in \(Fi_{24}\), which is a nonsplit extension of \(29:14\).
gap> SigmaFromMaxes( CharacterTable( "Fi24'.2" ), "29AB", > [ CharacterTable( "29:28" ) ], [ 1 ], "outer" ); 0
In the case \(S = Fi_{22}\), there are four classes of maximal subgroups that contain \(s\) of order \(16\). They extend to \(G = Fi_{22}.2\), and none of the novelties in \(G\) (i. e., subgroups of \(G\) that are maximal in \(G\) but whose intersections with \(S\) are not maximal in \(S\)) contains \(s\), cf. [CCN+85, p. 163].
gap> 16 in OrdersClassRepresentatives( CharacterTable( "U4(2).2" ) ); false gap> 16 in OrdersClassRepresentatives( CharacterTable( "G2(3).2" ) ); false
The character tables of three of the four extensions are available in the GAP Character Table Library. The permutation character on the cosets of the fourth extension can be obtained as the extension of the permutation character of \(S\) on the cosets of its maximal subgroup of the type \(2^{5+8}:(S_3 \times A_6)\).
gap> t2:= CharacterTable( "Fi22.2" );; gap> prim:= List( [ "Fi22.2M4", "(2x2^(1+8)):(U4(2):2x2)", "2F4(2)" ], > n -> PossiblePermutationCharacters( CharacterTable( n ), t2 ) );; gap> t:= CharacterTable( "Fi22" );; gap> pi:= PossiblePermutationCharacters( > CharacterTable( "2^(5+8):(S3xA6)" ), t ); [ Character( CharacterTable( "Fi22" ), [ 3648645, 56133, 10629, 2245, 567, 729, 405, 81, 549, 165, 133, 37, 69, 20, 27, 81, 9, 39, 81, 19, 1, 13, 33, 13, 1, 0, 13, 13, 5, 1, 0, 0, 0, 8, 4, 0, 0, 9, 3, 15, 3, 1, 1, 1, 1, 3, 3, 1, 0, 0, 0, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2 ] ) ] gap> torso:= CompositionMaps( pi[1], InverseMap( GetFusionMap( t, t2 ) ) ); [ 3648645, 56133, 10629, 2245, 567, 729, 405, 81, 549, 165, 133, 37, 69, 20, 27, 81, 9, 39, 81, 19, 1, 13, 33, 13, 1, 0, 13, 13, 5, 1, 0, 0, 0, 8, 4, 0, 9, 3, 15, 3, 1, 1, 1, 3, 3, 1, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 1, 1, 2 ] gap> ext:= PermChars( t2, rec( torso:= torso ) );; gap> Add( prim, ext ); gap> prim:= Concatenation( prim );; Length( prim ); 4 gap> spos:= Position( OrdersClassRepresentatives( t2 ), 16 );; gap> List( prim, x -> x[ spos ] ); [ 1, 1, 4, 1 ] gap> sigma:= ApproxP( prim, spos );; gap> Maximum( sigma{ Difference( PositionsProperty( > OrdersClassRepresentatives( t2 ), IsPrimeInt ), > ClassPositionsOfDerivedSubgroup( t2 ) ) } ); 251/3861
In the case \(S = HN\), the unique maximal subgroup \(U_3(8).3\) that contains the fixed element \(s\) of order \(19\) extends to a group of the type \(U_3(8).6\) in \(HN.2\).
gap> SigmaFromMaxes( CharacterTable( "HN.2" ), "19AB", > [ CharacterTable( "U3(8).6" ) ], [ 1 ], "outer" ); 1/6875
In the case \(S = HS\), there are two classes of maximal subgroups that contain \(s\) of order \(15\). They extend to \(G = HS.2\), and none of the novelties in \(G\) contains \(s\) (cf. [CCN+85, p. 80]).
gap> SigmaFromMaxes( CharacterTable( "HS.2" ), "15A", > [ CharacterTable( "S8x2" ), > CharacterTable( "5:4" ) * CharacterTable( "A5.2" ) ], [ 1, 1 ], > "outer" ); 36/275
In the case \(S = He\), there are three classes of maximal subgroups that contain \(s\) in the class 14C. They extend to \(G = He.2\), and none of the novelties in \(G\) contains \(s\) (cf. [CCN+85, p. 104]). We compute the extensions of the corresponding primitive permutation characters of \(S\).
gap> t:= CharacterTable( "He" );; gap> t2:= CharacterTable( "He.2" );; gap> prim:= PrimitivePermutationCharacters( t );; gap> spos:= Position( AtlasClassNames( t ), "14C" );; gap> prim:= Filtered( prim, x -> x[ spos ] <> 0 );; gap> map:= InverseMap( GetFusionMap( t, t2 ) );; gap> torso:= List( prim, pi -> CompositionMaps( pi, map ) ); [ [ 187425, 945, 449, 0, 21, 21, 25, 25, 0, 0, 5, 0, 0, 7, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0 ], [ 244800, 0, 64, 0, 84, 0, 0, 16, 0, 0, 4, 24, 45, 3, 4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 ], [ 652800, 0, 512, 120, 72, 0, 0, 0, 0, 0, 8, 8, 22, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 2, 0 ] ] gap> ext:= List( torso, x -> PermChars( t2, rec( torso:= x ) ) ); [ [ Character( CharacterTable( "He.2" ), [ 187425, 945, 449, 0, 21, 21, 25, 25, 0, 0, 5, 0, 0, 7, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 315, 15, 0, 0, 3, 7, 7, 3, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0 ] ) ], [ Character( CharacterTable( "He.2" ), [ 244800, 0, 64, 0, 84, 0, 0, 16, 0, 0, 4, 24, 45, 3, 4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 360, 0, 0, 0, 6, 0, 0, 0, 0, 0, 3, 2, 2, 0, 0, 0, 0, 0, 0 ] ) ], [ Character( CharacterTable( "He.2" ), [ 652800, 0, 512, 120, 72, 0, 0, 0, 0, 0, 8, 8, 22, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 2, 0, 480, 0, 120, 0, 12, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 1, 1 ] ) ] ] gap> spos:= Position( AtlasClassNames( t2 ), "14CD" );; gap> sigma:= ApproxP( Concatenation( ext ), spos );; gap> Maximum( sigma{ Difference( PositionsProperty( > OrdersClassRepresentatives( t2 ), IsPrimeInt ), > ClassPositionsOfDerivedSubgroup( t2 ) ) } ); 37/9520
In the case \(S = O'N\), the two classes of maximal subgroups of the type \(L_2(31)\) do not extend to \(G = O'N.2\), and a class of novelties of the structure \(31:30\) appears (see [CCN+85, p. 132]).
gap> SigmaFromMaxes( CharacterTable( "ON.2" ), "31AB", > [ CharacterTable( "P:Q", [ 31, 30 ] ) ], [ 1 ], "outer" ); 1/84672
Now we consider also \(σ(G,\hat{s})\), for suitable \(\hat{s} \in G \setminus S\); this yields lower bounds for the spread of the nonsimple groups \(G\). (These results are shown in the last two columns of [BGK08, Table 9].)
As above, we use the known character tables of the maximal subgroups in order to compute the optimal choice for \(\hat{s} \in G \setminus S\). (We may use the function ProbGenInfoSimple although the groups are not simple; all we need is that the relevant maximal subgroups are self-normalizing.)
gap> sporautinfo2:= [];; gap> for name in List( sporautinfo, x -> x[1] ) do > Add( sporautinfo2, ProbGenInfoSimple( CharacterTable( name ) ) ); > od; gap> PrintFormattedArray( sporautinfo2 ); F3+.2 19/5684 299 [ "42E" ] [ 10 ] Fi22.2 1165/20592 17 [ "24G" ] [ 3 ] HN.2 1/1425 1424 [ "24B" ] [ 4 ] HS.2 21/550 26 [ "20C" ] [ 4 ] He.2 33/4165 126 [ "24A" ] [ 2 ] J2.2 1/15 14 [ "14A" ] [ 1 ] J3.2 77/10260 133 [ "34A" ] [ 1 ] M12.2 113/495 4 [ "12B" ] [ 3 ] M22.2 8/33 4 [ "10A" ] [ 4 ] McL.2 1/135 134 [ "22A" ] [ 1 ] ON.2 61/109368 1792 [ "22A", "38A" ] [ 1, 1 ] Suz.2 1/351 350 [ "28A" ] [ 1 ] gap> for entry in sporautinfo2 do > DisplayProbGenMaxesInfo( CharacterTable( entry[1] ), entry[4] ); > od; F3+.2, 42E: 2^12.M24 (2) 2^2.U6(2):S3x2 (1) 2^(3+12).(L3(2)xS6) (2) (S3xS3xG2(3)):2 (1) S6xL2(8):3 (1) 7:6xS7 (1) 7^(1+2)_+:(6xS3).2 (2) Fi22.2, 24G: Fi22.2M4 (1) 2^(5+8):(S3xS6) (1) 3^5:(2xU4(2).2) (1) HN.2, 24B: 2^(1+8)_+.(A5xA5).2^2 (1) 5^2.5.5^2.4S5 (2) HN.2M13 (1) HS.2, 20C: (2xA6.2^2).2 (1) HS.2N5 (2) 5:4xS5 (1) He.2, 24A: 2^(1+6)_+.L3(2).2 (1) S4xL3(2).2 (1) J2.2, 14A: L3(2).2x2 (1) J3.2, 34A: L2(17)x2 (1) M12.2, 12B: L2(11).2 (1) 2^3.(S4x2) (1) 3^(1+2):D8 (1) M22.2, 10A: M22.2M4 (1) A6.2^2 (1) L2(11).2 (2) McL.2, 22A: 2xM11 (1) ON.2, 22A: J1x2 (1) ON.2, 38A: J1x2 (1) Suz.2, 28A: (A4xL3(4):2_3):2 (1)
In the other six cases, we do not have the complete lists of primitive permutation characters, so we choose a suitable element \(\hat{s}\) for each group. It is sufficient to prescribe \(|\hat{s}|\), as follows.
gap> sporautchoices:= [ > [ "Fi22", "Fi22.2", 42 ], > [ "Fi24'", "Fi24'.2", 46 ], > [ "He", "He.2", 42 ], > [ "HN", "HN.2", 44 ], > [ "HS", "HS.2", 30 ], > [ "ON", "ON.2", 38 ], ];;
First we list the maximal subgroups of the corresponding simple groups that contain the square of \(\hat{s}\).
gap> for triple in sporautchoices do > tbl:= CharacterTable( triple[1] ); > tbl2:= CharacterTable( triple[2] ); > spos2:= PowerMap( tbl2, 2, > Position( OrdersClassRepresentatives( tbl2 ), triple[3] ) ); > spos:= Position( GetFusionMap( tbl, tbl2 ), spos2 ); > DisplayProbGenMaxesInfo( tbl, AtlasClassNames( tbl ){ [ spos ] } ); > od; Fi22, 21A: O8+(2).3.2 (1) S3xU4(3).2_2 (1) A10.2 (1) A10.2 (1) F3+, 23A: Fi23 (1) F3+M7 (1) He, 21B: 3.A7.2 (1) 7^(1+2):(S3x3) (1) 7:3xpsl(3,2) (2) HN, 22A: 2.HS.2 (1) HS, 15A: A8.2 (1) 5:4xa5 (1) ON, 19B: L3(7).2 (1) ONM2 (1) J1 (1)
According to [CCN+85], exactly the following maximal subgroups of the simple group \(S\) in the above list do not extend to \(Aut(S)\): The two \(S_{10}\) type subgroups of \(Fi_{22}\) and the two \(L_3(7).2\) type subgroups of \(O'N\).
Furthermore, the following maximal subgroups of \(Aut(S)\) with the property that the intersection with \(S\) is not maximal in \(S\) have to be considered whether they contain \(s^{\prime}\): \(G_2(3).2\) and \(3^5:(2 \times U_4(2).2)\) in \(Fi_{22}.2\). (Note that the order of the \(7^{1+2}_+:(3 \times D_{16})\) type subgroup in \(O'N.2\) is obviously not divisible by \(19\).)
gap> 42 in OrdersClassRepresentatives( CharacterTable( "G2(3).2" ) ); false gap> Size( CharacterTable( "U4(2)" ) ) mod 7 = 0; false
So we take the extensions of the above maximal subgroups, as described in [CCN+85].
gap> SigmaFromMaxes( CharacterTable( "Fi22.2" ), "42A", > [ CharacterTable( "O8+(2).3.2" ) * CharacterTable( "Cyclic", 2 ), > CharacterTable( "S3" ) * CharacterTable( "U4(3).(2^2)_{122}" ) ], > [ 1, 1 ] ); 163/1170 gap> SigmaFromMaxes( CharacterTable( "Fi24'.2" ), "46A", > [ CharacterTable( "Fi23" ) * CharacterTable( "Cyclic", 2 ), > CharacterTable( "2^12.M24" ) ], > [ 1, 1 ] ); 566/5481 gap> SigmaFromMaxes( CharacterTable( "He.2" ), "42A", > [ CharacterTable( "3.A7.2" ) * CharacterTable( "Cyclic", 2 ), > CharacterTable( "7^(1+2):(S3x6)" ), > CharacterTable( "7:6" ) * CharacterTable( "L3(2)" ) ], > [ 1, 1, 1 ] ); 1/119 gap> SigmaFromMaxes( CharacterTable( "HN.2" ), "44A", > [ CharacterTable( "4.HS.2" ) ], > [ 1 ] ); 997/192375 gap> SigmaFromMaxes( CharacterTable( "HS.2" ), "30A", > [ CharacterTable( "S8" ) * CharacterTable( "C2" ), > CharacterTable( "5:4" ) * CharacterTable( "S5" ) ], > [ 1, 1 ] ); 36/275 gap> SigmaFromMaxes( CharacterTable( "ON.2" ), "38A", > [ CharacterTable( "J1" ) * CharacterTable( "C2" ) ], > [ 1 ] ); 61/109368
We are interested in simple groups \(G\) for which ProbGenInfoSimple does not guarantee \(𝕊/~(G) \geq 3\). So we examine the remaining tables of simple groups in the GAP Character Table Library, and distinguish the following three cases: Either ProbGenInfoSimple yields the lower bound at least three, or a smaller bound, or the computation of a lower bound fails because not enough information is available to compute the primitive permutation characters.
gap> names:= AllCharacterTableNames( IsSimple, true, IsAbelian, false, > IsDuplicateTable, false );; gap> names:= Difference( names, spornames );; gap> fails:= [];; gap> lessthan3:= [];; gap> atleast3:= [];; gap> for name in names do > tbl:= CharacterTable( name ); > info:= ProbGenInfoSimple( tbl ); > if info = fail then > Add( fails, name ); > elif info[3] < 3 then > Add( lessthan3, info ); > else > Add( atleast3, info ); > fi; > od;
For the following simple groups, (currently) not enough information is available in the GAP Character Table Library and in the GAP Library of Tables of Marks, for computing a lower bound for \(σ(G)\). Some of these groups will be dealt with in later sections, and for the other groups, the bounds derived with theoretical arguments in [BGK08] are sufficient, so we need no GAP computations for them.
gap> fails; [ "2E6(2)", "2F4(8)", "3D4(3)", "3D4(4)", "A14", "A15", "A16", "A17", "A18", "A19", "E6(2)", "L4(4)", "L4(5)", "L4(9)", "L5(3)", "L8(2)", "O10+(2)", "O10+(3)", "O10-(2)", "O10-(3)", "O12+(2)", "O12+(3)", "O12-(2)", "O12-(3)", "O7(5)", "O8+(7)", "O8-(3)", "O9(3)", "R(27)", "S10(2)", "S12(2)", "S4(7)", "S4(8)", "S4(9)", "S6(4)", "S6(5)", "S8(3)", "U4(4)", "U4(5)", "U5(3)", "U5(4)", "U6(4)", "U7(2)" ]
The following simple groups appear in [BGK08, Table 1–6]. More detailed computations can be found in the sections 11.5-2, 11.5-3, 11.5-4, 11.5-12, 11.5-13, 11.5-20, 11.5-23, 11.5-24.
gap> PrintFormattedArray( lessthan3 ); A5 1/3 2 [ "5A" ] [ 1 ] A6 2/3 1 [ "5A" ] [ 2 ] A7 2/5 2 [ "7A" ] [ 2 ] O7(3) 199/351 1 [ "14A" ] [ 3 ] O8+(2) 334/315 0 [ "15A", "15B", "15C" ] [ 7, 7, 7 ] O8+(3) 863/1820 2 [ "20A", "20B", "20C" ] [ 8, 8, 8 ] S6(2) 4/7 1 [ "9A" ] [ 4 ] S8(2) 8/15 1 [ "17A" ] [ 3 ] U4(2) 21/40 1 [ "12A" ] [ 2 ] U4(3) 53/135 2 [ "7A" ] [ 7 ]
For the following simple groups \(G\), the inequality \(σ(G) < 1/3\) follows from the loop above. The columns show the name of \(G\), the values \(σ(G)\) and \(𝕊/~(G)\), the class names of \(s\) for which these values are attained, and \(|𝕄(G,s)|\).
(We increase the line length for this table. Even with this width, the entry for the group \(L_7(2)\) would not fit on one screen line, we show it separately below.)
gap> oldsize:= SizeScreen();; gap> SizeScreen( [ 80 ] );; gap> PrintFormattedArray( Filtered( atleast3, l -> l[1] <> "L7(2)" ) ); 2F4(2)' 118/1755 14 [ "16A" ] [ 2 ] 3D4(2) 1/5292 5291 [ "13A" ] [ 1 ] A10 3/10 3 [ "21A" ] [ 1 ] A11 2/105 52 [ "11A" ] [ 2 ] A12 2/9 4 [ "35A" ] [ 1 ] A13 4/1155 288 [ "13A" ] [ 5 ] A8 3/14 4 [ "15A" ] [ 1 ] A9 9/35 3 [ "9A", "9B" ] [ 4, 4 ] F4(2) 9/595 66 [ "13A" ] [ 5 ] G2(3) 1/7 6 [ "13A" ] [ 3 ] G2(4) 1/21 20 [ "13A" ] [ 2 ] G2(5) 1/31 30 [ "7A", "21A" ] [ 10, 1 ] L2(101) 1/101 100 [ "51A", "17A" ] [ 1, 1 ] L2(103) 53/5253 99 [ "52A", "26A", "13A" ] [ 1, 1, 1 ] L2(107) 55/5671 103 [ "54A", "27A", "18A", "9A", "6A" ] [ 1, 1, 1, 1, 1 ] L2(109) 1/109 108 [ "55A", "11A" ] [ 1, 1 ] L2(11) 7/55 7 [ "6A" ] [ 1 ] L2(113) 1/113 112 [ "57A", "19A" ] [ 1, 1 ] L2(121) 1/121 120 [ "61A" ] [ 1 ] L2(125) 1/125 124 [ "63A", "21A", "9A", "7A" ] [ 1, 1, 1, 1 ] L2(13) 1/13 12 [ "7A" ] [ 1 ] L2(16) 1/15 14 [ "17A" ] [ 1 ] L2(17) 1/17 16 [ "9A" ] [ 1 ] L2(19) 11/171 15 [ "10A" ] [ 1 ] L2(23) 13/253 19 [ "6A", "12A" ] [ 1, 1 ] L2(25) 1/25 24 [ "13A" ] [ 1 ] L2(27) 5/117 23 [ "7A", "14A" ] [ 1, 1 ] L2(29) 1/29 28 [ "15A" ] [ 1 ] L2(31) 17/465 27 [ "8A", "16A" ] [ 1, 1 ] L2(32) 1/31 30 [ "3A", "11A", "33A" ] [ 1, 1, 1 ] L2(37) 1/37 36 [ "19A" ] [ 1 ] L2(41) 1/41 40 [ "21A", "7A" ] [ 1, 1 ] L2(43) 23/903 39 [ "22A", "11A" ] [ 1, 1 ] L2(47) 25/1081 43 [ "24A", "12A", "8A", "6A" ] [ 1, 1, 1, 1 ] L2(49) 1/49 48 [ "25A" ] [ 1 ] L2(53) 1/53 52 [ "27A", "9A" ] [ 1, 1 ] L2(59) 31/1711 55 [ "30A", "15A", "10A", "6A" ] [ 1, 1, 1, 1 ] L2(61) 1/61 60 [ "31A" ] [ 1 ] L2(64) 1/63 62 [ "65A", "13A" ] [ 1, 1 ] L2(67) 35/2211 63 [ "34A", "17A" ] [ 1, 1 ] L2(71) 37/2485 67 [ "36A", "18A", "12A", "9A", "6A" ] [ 1, 1, 1, 1, 1 ] L2(73) 1/73 72 [ "37A" ] [ 1 ] L2(79) 41/3081 75 [ "40A", "20A", "10A", "8A" ] [ 1, 1, 1, 1 ] L2(8) 1/7 6 [ "3A", "9A" ] [ 1, 1 ] L2(81) 1/81 80 [ "41A" ] [ 1 ] L2(83) 43/3403 79 [ "42A", "21A", "14A", "7A", "6A" ] [ 1, 1, 1, 1, 1 ] L2(89) 1/89 88 [ "45A", "15A", "9A" ] [ 1, 1, 1 ] L2(97) 1/97 96 [ "49A", "7A" ] [ 1, 1 ] L3(11) 1/6655 6654 [ "19A", "133A" ] [ 1, 1 ] L3(2) 1/4 3 [ "7A" ] [ 1 ] L3(3) 1/24 23 [ "13A" ] [ 1 ] L3(4) 1/5 4 [ "7A" ] [ 3 ] L3(5) 1/250 249 [ "31A" ] [ 1 ] L3(7) 1/1372 1371 [ "19A" ] [ 1 ] L3(8) 1/1792 1791 [ "73A" ] [ 1 ] L3(9) 1/2880 2879 [ "91A" ] [ 1 ] L4(3) 53/1053 19 [ "20A" ] [ 1 ] L5(2) 1/5376 5375 [ "31A" ] [ 1 ] L6(2) 365/55552 152 [ "21A", "63A" ] [ 2, 2 ] O8-(2) 1/63 62 [ "17A" ] [ 1 ] S4(4) 4/15 3 [ "17A" ] [ 2 ] S4(5) 1/5 4 [ "13A" ] [ 1 ] S6(3) 1/117 116 [ "14A" ] [ 2 ] Sz(32) 1/1271 1270 [ "5A", "25A" ] [ 1, 1 ] Sz(8) 1/91 90 [ "5A" ] [ 1 ] U3(11) 1/6655 6654 [ "37A" ] [ 1 ] U3(3) 16/63 3 [ "6A", "12A" ] [ 2, 2 ] U3(4) 1/160 159 [ "13A" ] [ 1 ] U3(5) 46/525 11 [ "10A" ] [ 2 ] U3(7) 1/1372 1371 [ "43A" ] [ 1 ] U3(8) 1/1792 1791 [ "19A" ] [ 1 ] U3(9) 1/3600 3599 [ "73A" ] [ 1 ] U5(2) 1/54 53 [ "11A" ] [ 1 ] U6(2) 5/21 4 [ "11A" ] [ 4 ] gap> SizeScreen( oldsize );; gap> First( atleast3, l -> l[1] = "L7(2)" ); [ "L7(2)", 1/4388290560, 4388290559, [ "127A" ], [ 1 ] ]
It should be mentioned that [BW75] states the following lower bounds for the uniform spread of the groups \(L_2(q)\).
| \(q-2\) | if \(4 \leq q\) is even, |
| \(q-1\) | if \(11 \leq q \equiv 1 \pmod{4}\), |
| \(q-4\) | if \(11 \leq q \equiv -1 \pmod{4}\). |
These bounds appear in the third column of the above table. Furthermore, [BW75] states that the (uniform) spread of alternating groups of even degree at least \(8\) is exactly \(4\).
For the sake of completeness, Table II gives an overview of the sets \(𝕄(G,s)\) for those cases in the above list that are needed in [BGK08] but that do not require a further discussion here. The structure of the maximal subgroups and the order of \(s\) in the table refer to the matrix groups not to the simple groups. The number of the subgroups has been shown above, the structure follows from [CCN+85].
| \(G\) | \(𝕄(G,s)\) | \(|s|\) | see [CCN+85] |
| \(SL(3,4) = 3.L_3(4)\) | \(3 \times L_3(2), 3 \times L_3(2), 3 \times L_3(2)\) | \(21\) | p. 23 |
| \(\Omega^-(8,2) = O^-_8(2)\) | \(\Omega^-(4,4).2 = L_2(16).2\) | \(17\) | p. 89 |
| \(Sp(4,4) = S_4(4)\) | \(\Omega^-(4,4).2 = L_2(16).2, Sp(2,16).2 = L_2(16).2\) | \(17\) | p. 44 |
| \(Sp(6,3) = 2.S_6(3)\) | \((4 \times U_3(3)).2, Sp(2,17).3 = 2.L_2(27).3\) | \(28\) | p. 113 |
| \(SU(3,3) = U_3(3)\) | \(3^{1+2}_+:8, GU(2,3) = 4.S_4\) | \(6\) | p. 14 |
| \(SU(3,5) = 3.U_3(5)\) | \(3 \times 5^{1+2}_+:8, GU(2,5) = 3 \times 2S_5\) | \(30\) | p. 34 |
| \(SU(5,2) = U_5(2)\) | \(L_2(11)\) | \(11\) | p. 73 |
We deal with automorphic extensions of those simple groups that are listed in Table I and that have been treated successfully in Section 11.4-3.
For the following groups, ProbGenInfoAlmostSimple can be used because GAP can compute their primitive permutation characters.
gap> list:= [ > [ "A5", "A5.2" ], > [ "A6", "A6.2_1" ], > [ "A6", "A6.2_2" ], > [ "A6", "A6.2_3" ], > [ "A7", "A7.2" ], > [ "A8", "A8.2" ], > [ "A9", "A9.2" ], > [ "A11", "A11.2" ], > [ "L3(2)", "L3(2).2" ], > [ "L3(3)", "L3(3).2" ], > [ "L3(4)", "L3(4).2_1" ], > [ "L3(4)", "L3(4).2_2" ], > [ "L3(4)", "L3(4).2_3" ], > [ "L3(4)", "L3(4).3" ], > [ "S4(4)", "S4(4).2" ], > [ "U3(3)", "U3(3).2" ], > [ "U3(5)", "U3(5).2" ], > [ "U3(5)", "U3(5).3" ], > [ "U4(2)", "U4(2).2" ], > [ "U4(3)", "U4(3).2_1" ], > [ "U4(3)", "U4(3).2_3" ], > ];; gap> autinfo:= [];; gap> fails:= [];; gap> for pair in list do > tbl:= CharacterTable( pair[1] ); > tblG:= CharacterTable( pair[2] ); > info:= ProbGenInfoSimple( tbl ); > spos:= List( info[4], x -> Position( AtlasClassNames( tbl ), x ) ); > Add( autinfo, ProbGenInfoAlmostSimple( tbl, tblG, spos ) ); > od; gap> PrintFormattedArray( autinfo ); A5.2 0 [ "5AB" ] [ 1 ] A6.2_1 2/3 [ "5AB" ] [ 2 ] A6.2_2 1/6 [ "5A" ] [ 1 ] A6.2_3 0 [ "5AB" ] [ 1 ] A7.2 1/15 [ "7AB" ] [ 1 ] A8.2 13/28 [ "15AB" ] [ 1 ] A9.2 1/4 [ "9AB" ] [ 1 ] A11.2 1/945 [ "11AB" ] [ 1 ] L3(2).2 1/4 [ "7AB" ] [ 1 ] L3(3).2 1/18 [ "13AB" ] [ 1 ] L3(4).2_1 3/10 [ "7AB" ] [ 3 ] L3(4).2_2 11/60 [ "7A" ] [ 1 ] L3(4).2_3 1/12 [ "7AB" ] [ 1 ] L3(4).3 1/64 [ "7A" ] [ 1 ] S4(4).2 0 [ "17AB" ] [ 2 ] U3(3).2 2/7 [ "6A", "12AB" ] [ 2, 2 ] U3(5).2 2/21 [ "10A" ] [ 2 ] U3(5).3 46/525 [ "10A" ] [ 2 ] U4(2).2 16/45 [ "12AB" ] [ 2 ] U4(3).2_1 76/135 [ "7A" ] [ 3 ] U4(3).2_3 31/162 [ "7AB" ] [ 3 ]
We see that from this list, the two groups \(A_6.2_1 = S_6\) and \(U_4(3).2_1\) require further computations (see Sections 11.5-3 and 11.5-24, respectively) because the bound in the second column is larger than \(1/2\).
Also \(U_4(2)\) is not done by the above, because in [BGK08, Table 4], an element \(s\) of order \(9\) is chosen for the simple group, see Section 11.5-23.
Finally, we deal with automorphic extensions of the groups \(L_4(3)\), \(O_8^-(2)\), \(S_6(3)\), and \(U_5(2)\).
For \(S = L_4(3)\) and \(s \in S\) of order \(20\), we have \(𝕄(S,s) = \{ (4 \times A_6):2 \}\), the subgroup has index \(2\,106\), see [CCN+85, p. 69].
gap> t:= CharacterTable( "L4(3)" );; gap> prim:= PrimitivePermutationCharacters( t );; gap> spos:= Position( AtlasClassNames( t ), "20A" );; gap> prim:= Filtered( prim, x -> x[ spos ] <> 0 ); [ Character( CharacterTable( "L4(3)" ), [ 2106, 106, 42, 0, 27, 27, 0, 46, 6, 6, 1, 7, 7, 0, 3, 3, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1 ] ) ]
For the three automorphic extensions of the structure \(G = S.2\), we compute the extensions of the permutation character, and the bounds \(σ^{\prime}(G,s)\).
gap> for name in [ "L4(3).2_1", "L4(3).2_2", "L4(3).2_3" ] do > t2:= CharacterTable( name ); > map:= InverseMap( GetFusionMap( t, t2 ) ); > torso:= List( prim, pi -> CompositionMaps( pi, map ) ); > ext:= Concatenation( List( torso, > x -> PermChars( t2, rec( torso:= x ) ) ) ); > sigma:= ApproxP( ext, Position( OrdersClassRepresentatives( t2 ), 20 ) ); > max:= Maximum( sigma{ Difference( PositionsProperty( > OrdersClassRepresentatives( t2 ), IsPrimeInt ), > ClassPositionsOfDerivedSubgroup( t2 ) ) } ); > Print( name, ":\n", ext, "\n", max, "\n" ); > od; L4(3).2_1: [ Character( CharacterTable( "L4(3).2_1" ), [ 2106, 106, 42, 0, 27, 0, 46, 6, 6, 1, 7, 0, 3, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 4, 0, 0, 6, 6, 6, 6, 2, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1 ] ) ] 0 L4(3).2_2: [ Character( CharacterTable( "L4(3).2_2" ), [ 2106, 106, 42, 0, 27, 27, 0, 46, 6, 6, 1, 7, 7, 0, 3, 3, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 306, 306, 42, 6, 10, 10, 0, 0, 15, 15, 3, 3, 3, 3, 0, 0, 1, 1, 0, 1, 1, 0, 0 ] ) ] 17/117 L4(3).2_3: [ Character( CharacterTable( "L4(3).2_3" ), [ 2106, 106, 42, 0, 27, 0, 46, 6, 6, 1, 7, 0, 3, 0, 0, 1, 1, 0, 0, 0, 1, 36, 0, 0, 6, 6, 2, 2, 2, 1, 1, 0, 0, 0 ] ) ] 2/117
For \(S = O_8^-(2)\) and \(s \in S\) of order \(17\), we have \(𝕄(S,s) = \{ L_2(16).2 \}\), the subgroup extends to \(L_2(16).4\) in \(S.2\), see [CCN+85, p. 89]. This is a non-split extension, so \(σ^{\prime}(S.2,s) = 0\) holds.
gap> SigmaFromMaxes( CharacterTable( "O8-(2).2" ), "17AB", > [ CharacterTable( "L2(16).4" ) ], [ 1 ], "outer" ); 0
For \(S = S_6(3)\) and \(s \in S\) irreducible of order \(14\), we have \(𝕄(S,s) = \{ (2 \times U_3(3)).2, L_2(27).3 \}\). In \(G = S.2\), the subgroups extend to \((4 \times U_3(3)).2\) and \(L_2(27).6\), respectively, see [CCN+85, p. 113]. In order to show that \(σ^{\prime}(G,s) = 7/3240\) holds, we compute the primitive permutation characters of \(S\) (cf. Section 11.4-3) and the unique extensions to \(G\) of those which are nonzero on \(s\).
gap> t:= CharacterTable( "S6(3)" );; gap> t2:= CharacterTable( "S6(3).2" );; gap> prim:= PrimitivePermutationCharacters( t );; gap> spos:= Position( AtlasClassNames( t ), "14A" );; gap> prim:= Filtered( prim, x -> x[ spos ] <> 0 );; gap> map:= InverseMap( GetFusionMap( t, t2 ) );; gap> torso:= List( prim, pi -> CompositionMaps( pi, map ) );; gap> ext:= List( torso, pi -> PermChars( t2, rec( torso:= pi ) ) ); [ [ Character( CharacterTable( "S6(3).2" ), [ 155520, 0, 288, 0, 0, 0, 216, 54, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 1, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 144, 288, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0 ] ) ], [ Character( CharacterTable( "S6(3).2" ), [ 189540, 1620, 568, 0, 486, 0, 0, 27, 540, 84, 24, 0, 0, 0, 0, 0, 54, 0, 0, 10, 0, 7, 1, 6, 6, 0, 0, 0, 0, 0, 0, 18, 0, 0, 0, 6, 12, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 234, 64, 30, 8, 0, 3, 90, 6, 0, 4, 10, 6, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0 ] ) ] ] gap> spos:= Position( AtlasClassNames( t2 ), "14A" );; gap> sigma:= ApproxP( Concatenation( ext ), spos );; gap> Maximum( sigma{ Difference( > PositionsProperty( OrdersClassRepresentatives( t2 ), IsPrimeInt ), > ClassPositionsOfDerivedSubgroup( t2 ) ) } ); 7/3240
For \(S = U_5(2)\) and \(s \in S\) of order \(11\), we have \(𝕄(S,s) = \{ L_2(11) \}\), the subgroup extends to \(L_2(11).2\) in \(S.2\), see [CCN+85, p. 73].
gap> SigmaFromMaxes( CharacterTable( "U5(2).2" ), "11AB", > [ CharacterTable( "L2(11).2" ) ], [ 1 ], "outer" ); 1/288
Here we clean the workspace for the first time. This may save more than \(100\) megabytes, due to the fact that the caches for tables of marks and character tables are flushed.
gap> CleanWorkspace();
We show that \(S = O_8^-(3) = \Omega^-(8,3)\) satisfies the following.
For \(s \in S\) of order \(41\), \(𝕄(S,s)\) consists of one group of the type \(L_2(81).2_1 = \Omega^-(4,9).2\).
\(σ(S,s) = 1/567\).
The only maximal subgroups of \(S\) containing elements of order \(41\) have the type \(L_2(81).2_1\), and there is one class of these subgroups, see [CCN+85, p. 141].
gap> SigmaFromMaxes( CharacterTable( "O8-(3)" ), "41A", > [ CharacterTable( "L2(81).2_1" ) ], [ 1 ] ); 1/567
We show that \(S = O_{10}^+(2) = \Omega^+(10,2)\) satisfies the following.
Ford\(s \in S\) of order \(45\), \(𝕄(S,s)\) consists of one group of the type \((A_5 \times U_4(2)).2 = (\Omega^-(4,2) \times \Omega^-(6,2)).2\).
\(σ(S,s) = 43/4\,216\).
For \(s\) as in (a), the maximal subgroup in (a) extends to \(S_5 \times U_4(2).2\) in \(G = Aut(S) = S.2\), and \(σ^{\prime}(G,s) = 23/248\).
The only maximal subgroups of \(S\) containing elements of order \(45\) are one class of groups \(H = (A_5 \times U_4(2)):2\), see [CCN+85, p. 146]. (Note that none of the groups \(S_8(2)\), \(O_8^+(2)\), \(L_5(2)\), \(O_8^-(2)\), and \(A_8\) contains elements of order \(45\).) \(H\) extends to subgroups of the type \(H.2 = S_5 \times U_4(2):2\) in \(G\), so we can compute \(1_H^S = (1_{H.2}^G)_S\).
gap> ForAny( [ "S8(2)", "O8+(2)", "L5(2)", "O8-(2)", "A8" ], > x -> 45 in OrdersClassRepresentatives( CharacterTable( x ) ) ); false gap> t:= CharacterTable( "O10+(2)" );; gap> t2:= CharacterTable( "O10+(2).2" );; gap> s2:= CharacterTable( "A5.2" ) * CharacterTable( "U4(2).2" ); CharacterTable( "A5.2xU4(2).2" ) gap> pi:= PossiblePermutationCharacters( s2, t2 );; gap> spos:= Position( OrdersClassRepresentatives( t2 ), 45 );; gap> approx:= ApproxP( pi, spos );; gap> Maximum( approx{ ClassPositionsOfDerivedSubgroup( t2 ) } ); 43/4216
Statement (c) follows from considering the outer classes of prime element order.
gap> Maximum( approx{ Difference( > PositionsProperty( OrdersClassRepresentatives( t2 ), IsPrimeInt ), > ClassPositionsOfDerivedSubgroup( t2 ) ) } ); 23/248
Alternatively, we can use SigmaFromMaxes.
gap> SigmaFromMaxes( t2, "45AB", [ s2 ], [ 1 ], "outer" ); 23/248
We show that \(S = O_{10}^-(2) = \Omega^-(10,2)\) satisfies the following.
For \(s \in S\) of order \(33\), \(𝕄(S,s)\) consists of one group of the type \(3 \times U_5(2) = GU(5,2)\).
\(σ(S,s) = 1/119\).
For \(s\) as in (a), the maximal subgroup in (a) extends to \((3 \times U_5(2)).2\) in \(G\), and \(σ^{\prime}(G,s) = 1/595\).
The only maximal subgroups of \(S\) containing elements of order \(11\) have the types \(A_{12}\) and \(3 \times U_5(2)\), see [CCN+85, p. 147]. So \(3 \times U_5(2)\) is the unique class of subgroups containing elements of order \(33\). This shows statement (a), and statement (b) follows using SigmaFromMaxes.
gap> SigmaFromMaxes( CharacterTable( "O10-(2)" ), "33A", > [ CharacterTable( "Cyclic", 3 ) * CharacterTable( "U5(2)" ) ], [ 1 ] ); 1/119
The structure of the maximal subgroup of \(G\) follows from [CCN+85, p. 147]. We create its character table with a generic construction that is based on the fact that the outer automorphism acts nontrivially on the two direct factors; this determines the character table uniquely. (See [Brec] for details.)
gap> tblG:= CharacterTable( "U5(2)" );; gap> tblMG:= CharacterTable( "Cyclic", 3 ) * tblG;; gap> tblGA:= CharacterTable( "U5(2).2" );; gap> acts:= PossibleActionsForTypeMGA( tblMG, tblG, tblGA );; gap> poss:= Concatenation( List( acts, pi -> > PossibleCharacterTablesOfTypeMGA( tblMG, tblG, tblGA, pi, > "(3xU5(2)).2" ) ) ); [ rec( MGfusMGA := [ 1, 2, 3, 4, 4, 5, 5, 6, 7, 8, 9, 10, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 17, 17, 18, 18, 19, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 31, 32, 33, 35, 34, 37, 36, 38, 39, 40, 41, 42, 43, 45, 44, 47, 46, 49, 48, 51, 50, 52, 54, 53, 56, 55, 57, 58, 60, 59, 62, 61, 64, 63, 66, 65, 68, 67, 69, 71, 70, 73, 72, 75, 74, 77, 76 ], table := CharacterTable( "(3xU5(2)).2" ) ) ]
Now statement (c) follows using SigmaFromMaxes.
gap> SigmaFromMaxes( CharacterTable( "O10-(2).2" ), "33AB", > [ poss[1].table ], [ 1 ], "outer" ); 1/595
We show that \(S = O_{12}^+(2) = \Omega^+(12,2)\) satisfies the following.
For \(s \in S\) of the type \(4^- \perp 8^-\) (i. e., \(s\) decomposes the natural \(12\)-dimensional module for \(GO^+_{12}(2) = S.2\) into an orthogonal sum of two irreducible modules of the dimensions \(4\) and \(8\), respectively) and of order \(85\), \(𝕄(S,s)\) consists of one group of the type \(G_8 = (\Omega^-(4,2) \times \Omega^-(8,2)).2\) and two groups of the type \(L_4(4).2^2 = \Omega^+(6,4).2^2\) that are conjugate in \(G = Aut(S) = S.2 = SO^+(12,2)\) but not conjugate in \(S\).
\(σ(S,s) = 7\,675/1\,031\,184\).
\(σ^{\prime}(G,s) = 73/1\,008\).
The element \(s\) is a ppd\((12,2;8)\)-element in the sense of [GPPS99], so the maximal subgroups of \(S\) that contain \(s\) are among the nine cases (2.1)–(2.9) listed in this paper; in the notation of this paper, we have \(q = 2\), \(d = 12\), \(e = 8\), and \(r = 17\). Case (2.1) does not occur for orthogonal groups and \(q = 2\), according to [KL90]; case (2.2) contributes a unique maximal subgroup, the stabilizer \(G_8\) of the orthogonal decomposition; the cases (2.3), (2.4) (a), (2.5), and (2.6) (a) do not occur because \(r ≠ e+1\) in our situation; case (2.4) (b) describes extension field type subgroups that are contained in \(ΓL(6,4)\), which yields the candidates \(GU(6,2).2 \cong 3.U_6(2).S_3\) –but \(3.U_6(2).3\) does not contain elements of order \(85\)– and \(\Omega^+(6,4).2^2 \cong L_4(4).2^2\) (two classes by [KL90, Prop. 4.3.14]); the cases (2.6) (b)–(c) and (2.8) do not occur because they require \(d \leq 8\); case (2.7) does not occur because [GPPS99, Table 5] contains no entry for \(r = 2e+1 = 17\); finally, case (2.9) does not occur because it requires \(e \in \{ d-1, d \}\) in the case \(r = 2e+1\).
So we need the permutation characters of the actions on the cosets of \(L_4(4).2^2\) (two classes) and \(G_8\). According to [KL90, Prop. 4.1.6], \(G_8\) has the structure \((\Omega^-(4,2) \times \Omega^-(8,2)).2\).
Newer versions of the GAP Character Table Library contain the character table of \(S\), but it is still easier to work with the table of \(G\), which was already available at the times when the first version of these examples was created.
The two classes of \(L_4(4).2^2\) type subgroups in \(S\) are fused in \(G\). This can be seen from the fact that inducing the trivial character of a subgroup \(H_1 = L_4(4).2^2\) of \(S\) to \(G\) yields a character \(\psi\) whose values are not all even; note that if \(H_1\) would extend in \(G\) to a subgroup of twice the size of \(H_1\) then \(\psi\) would be induced from a degree two character of this subgroup whose values are all even, and induction preserves this property.
gap> t:= CharacterTable( "O12+(2).2" );; gap> h1:= CharacterTable( "L4(4).2^2" );; gap> psi:= PossiblePermutationCharacters( h1, t );; gap> Length( psi ); 1 gap> ForAny( psi[1], IsOddInt ); true
The fixed element \(s\) of order \(85\) is contained in a member of each of the two conjugacy classes of the type \(L_4(4).2^2\) in \(S\), since \(S\) contains only one class of subgroups of the order \(85\); note that the order of the Sylow \(17\) centralizer (in both \(S\) and \(G\)) is not divisible by \(25\).
gap> SizesCentralizers( t ){ PositionsProperty( > OrdersClassRepresentatives( t ), x -> x = 17 ) } / 25; [ 408/5, 408/5 ]
This implies that the restriction of \(\psi\) to \(S\) is the sum \(\psi_S = \pi_1 + \pi_2\), say, of the first two interesting permutation characters of \(S\).
The subgroup \(G_8\) of \(S\) extends to a group of the structure \(H_2 = \Omega^-(4,2).2 \times \Omega^-(8,2).2\) in \(G\), inducing the trivial characters of \(H_2\) to \(G\) yields a permutation character \(\varphi\) of \(G\) whose restriction to \(S\) is the third permutation character \(\varphi_S = \pi_3\), say.
gap> h2:= CharacterTable( "S5" ) * CharacterTable( "O8-(2).2" );; gap> phi:= PossiblePermutationCharacters( h2, t );; gap> Length( phi ); 1
We have \(\pi_1(1) = \pi_2(1)\) and \(\pi_1(s) = \pi_2(s)\), the latter again because \(S\) contains only one class of subgroups of order \(85\).
Now statement (a) follows from the fact that \(\pi_i(s) = 1\) holds for \(1 \leq i \leq 3\).
gap> prim:= Concatenation( psi, phi );; gap> spos:= Position( OrdersClassRepresentatives( t ), 85 ); 213 gap> List( prim, x -> x[ spos ] ); [ 2, 1 ]
For statement (b), we compute \(σ(S,s)\). Note that we have to consider only classes inside \(S = G^{\prime}\), and that
\[ σ( g, s ) = \sum_{i=1}^3 \frac{\pi_i(s) \cdot \pi_i(g)}{\pi_i(1)} = \frac{\psi(s) \cdot \psi(g)}{\psi(1)} + \frac{\varphi(s) \cdot \varphi(g)}{\varphi(1)} \]
holds for \(g \in S^{\times}\), so the characters \(\psi\) and \(\varphi\) are sufficient.
gap> approx:= ApproxP( prim, spos );; gap> Maximum( approx{ ClassPositionsOfDerivedSubgroup( t ) } ); 7675/1031184
Statement (c) follows from considering the outer involution classes. Note that by [BGK08, Remark after Proposition 5.14], only the subgroup \(H_2\) need to be considered, no novelties appear.
gap> Maximum( approx{ Difference( > PositionsProperty( OrdersClassRepresentatives( t ), IsPrimeInt ), > ClassPositionsOfDerivedSubgroup( t ) ) } ); 73/1008
We show that \(S = O_{12}^-(2) = \Omega^-(12,2)\) satisfies the following.
For \(s \in S\) irreducible of order \(2^6+1 = 65\), \(𝕄(S,s)\) consists of two groups of the types \(U_4(4).2 = \Omega^-(6,4).2\) and \(L_2(64).3 = \Omega^-(4,8).3\), respectively.
\(σ(S,s) = 1/1\,023\).
\(σ^{\prime}(Aut(S),s) = 1/347\,820\).
By [Ber00], \(𝕄(S,s)\) consists of extension field subgroups, which have the structures \(U_4(4).2\) and \(L_2(64).3\), respectively, and by [KL90, Prop. 4.3.16], there is just one class of each of these types.
Newer versions of the GAP Character Table Library contain the character table of \(S\), but using this table for the computations is not easier than using the table of \(G = Aut(S) = O_{12}^-(2).2\), which was already available at the times when the first version of these examples was created. So we compute the permutation characters \(\pi_1, \pi_2\) of the extensions of the groups in \(𝕄(S,s)\) to \(G\) –these maximal subgroups have the structures \(U_4(4).4\) and \(L_2(64).6\), respectively– and compute the fixed point ratios of the restrictions to \(S\).
gap> t:= CharacterTable( "O12-(2).2" );; gap> s1:= CharacterTable( "U4(4).4" );; gap> pi1:= PossiblePermutationCharacters( s1, t );; gap> s2:= CharacterTable( "L2(64).6" );; gap> pi2:= PossiblePermutationCharacters( s2, t );; gap> prim:= Concatenation( pi1, pi2 );; Length( prim ); 2
Now statement (a) follows from the fact that \(\pi_1(s) = \pi_2(s) = 1\) holds.
gap> spos:= Position( OrdersClassRepresentatives( t ), 65 );; gap> List( prim, x -> x[ spos ] ); [ 1, 1 ]
For statement (b), we compute \(σ(S,s)\); note that we have to consider only classes inside \(S = G^{\prime}\).
gap> approx:= ApproxP( prim, spos );; gap> Maximum( approx{ ClassPositionsOfDerivedSubgroup( t ) } ); 1/1023
Statement (c) follows from the values on the outer involution classes.
gap> Maximum( approx{ Difference( > PositionsProperty( OrdersClassRepresentatives( t ), IsPrimeInt ), > ClassPositionsOfDerivedSubgroup( t ) ) } ); 1/347820
We show that \(S = S_6(4) = Sp(6,4)\) satisfies the following.
For \(s \in S\) irreducible of order \(65\), \(𝕄(S,s)\) consists of two groups of the types \(U_4(4).2 = \Omega^-(6,4).2\) and \(L_2(64).3 = Sp(2,64).3\), respectively.
\(σ(S,s) = 16/63\).
\(σ^{\prime}(Aut(S),s) = 0\).
By [Ber00], the element \(s\) is contained in maximal subgroups of the given types, and by [KL90, Prop. 4.3.10, 4.8.6], there is exactly one class of these subgroups.
The character tables of these two subgroups are currently not contained in the GAP Character Table Library. We compute the permutation character induced from the first subgroup as the unique character of the right degree that is combinatorially possible (cf. [BP98]).
gap> t:= CharacterTable( "S6(4)" );; gap> degree:= Size( t ) / ( 2 * Size( CharacterTable( "U4(4)" ) ) );; gap> pi1:= PermChars( t, rec( torso:= [ degree ] ) );; gap> Length( pi1 ); 1
The index of the second subgroup is too large for this simpleminded approach; therefore, we first restrict the set of possible irreducible constituents of the permutation character to those of \(1_H^G\), where \(H\) is the derived subgroup of \(L_2(64).3\), for which the character table is available.
gap> CharacterTable( "L2(64).3" ); CharacterTable( "U4(4).2" ); fail fail gap> s:= CharacterTable( "L2(64)" );; gap> subpi:= PossiblePermutationCharacters( s, t );; gap> Length( subpi ); 1 gap> scp:= MatScalarProducts( t, Irr( t ), subpi );; gap> nonzero:= PositionsProperty( scp[1], x -> x <> 0 ); [ 1, 11, 13, 14, 17, 18, 32, 33, 56, 58, 59, 73, 74, 77, 78, 79, 80, 93, 95, 96, 103, 116, 117, 119, 120 ] gap> const:= RationalizedMat( Irr( t ){ nonzero } );; gap> degree:= Size( t ) / ( 3 * Size( s ) ); 5222400 gap> pi2:= PermChars( t, rec( torso:= [ degree ], chars:= const ) );; gap> Length( pi2 ); 1 gap> prim:= Concatenation( pi1, pi2 );;
Now statement (a) follows from the fact that \(\pi_1(s) = \pi_2(s) = 1\) holds.
gap> spos:= Position( OrdersClassRepresentatives( t ), 65 );; gap> List( prim, x -> x[ spos ] ); [ 1, 1 ]
For statement (b), we compute \(σ(G,s)\).
gap> Maximum( ApproxP( prim, spos ) ); 16/63
In order to prove statement (c), we have to consider only the extensions of the above permutation characters of \(S\) to \(Aut(S) \cong S.2\) (cf. [BGK08, Section 2.2]).
gap> t2:= CharacterTable( "S6(4).2" );; gap> tfust2:= GetFusionMap( t, t2 );; gap> cand:= List( prim, x -> CompositionMaps( x, InverseMap( tfust2 ) ) );; gap> ext:= List( cand, pi -> PermChars( t2, rec( torso:= pi ) ) ); [ [ Character( CharacterTable( "S6(4).2" ), [ 2016, 512, 96, 128, 32, 120, 0, 6, 16, 40, 24, 0, 8, 136, 1, 6, 6, 1, 32, 0, 8, 6, 2, 0, 2, 0, 0, 4, 0, 16, 32, 1, 8, 2, 6, 2, 1, 2, 4, 0, 0, 1, 6, 0, 1, 10, 0, 1, 1, 0, 10, 10, 4, 0, 1, 0, 2, 0, 2, 1, 2, 2, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 32, 0, 0, 8, 0, 0, 0, 0, 8, 8, 0, 0, 0, 0, 8, 0, 0, 0, 2, 2, 0, 2, 2, 0, 2, 2, 2, 0, 0 ] ) ], [ Character( CharacterTable( "S6(4).2" ), [ 5222400, 0, 0, 0, 1280, 0, 960, 120, 0, 0, 0, 0, 0, 0, 1600, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 1, 0, 0, 15, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 960, 0, 0, 0, 16, 0, 24, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 1, 0, 0, 3, 0, 0, 0, 0, 0 ] ) ] ] gap> spos2:= Position( OrdersClassRepresentatives( t2 ), 65 );; gap> sigma:= ApproxP( Concatenation( ext ), spos2 );; gap> Maximum( approx{ Difference( > PositionsProperty( OrdersClassRepresentatives( t2 ), IsPrimeInt ), > ClassPositionsOfDerivedSubgroup( t2 ) ) } ); 0
For the simple group, we can alternatively consider a reducible element \(s: 2 \perp 4\) of order \(85\), which is a multiple of the primitive prime divisor \(r = 17\) of \(4^4-1\). So we have \(e = 4\), \(d = 6\), and \(q = 4\), in the terminology of [GPPS99]. Then \(𝕄(S,s)\) consists of two groups, of the types \(\Omega^+(6,4).2 \cong L_4(4).2_2\) and \(Sp(2,4) \times Sp(4,4)\). This can be shown by checking [GPPS99, Ex. 2.1–2.9]. Ex. 2.1 yields the candidates \(\Omega^{\pm}(6,4).2\), but only \(\Omega^+(6,4).2\) contains elements of order \(85\). Ex. 2.2 yields the stabilizer of a two-dimensional subspace, which has the structure \(Sp(2,4) \times Sp(4,4)\), by [KL90]. All other cases except Ex. 2.4 (b) are excluded by the fact that \(r = 4e+1\), and Ex. 2.4 (b) does not apply because \(d/\gcd(d,e)\) is odd.
gap> SigmaFromMaxes( CharacterTable( "S6(4)" ), "85A", > [ CharacterTable( "L4(4).2_2" ), > CharacterTable( "A5" ) * CharacterTable( "S4(4)" ) ], [ 1, 1 ] ); 142/455
This bound is not as good as the one obtained from the irreducible element of order \(65\) used above.
gap> 16/63 < 142/455; true
We show that \(S = S_6(5) = PSp(6,5)\) satisfies the following.
For \(s \in S\) of the type \(2 \perp 4\) (i. e., the preimage of \(s\) in \(Sp(6,5) = 2.G\) decomposes the natural \(6\)-dimensional module for \(Sp(6,5)\) into an orthogonal sum of two irreducible modules of the dimensions \(2\) and \(4\), respectively) and of order \(78\), \(𝕄(S,s)\) consists of one group of the type \(G_2 = 2.(PSp(2,5) \times PSp(4,5))\).
\(σ(S,s) = 9/217\).
The order of \(s\) is a multiple of the primitive prime divisor \(r = 13\) of \(5^4-1\), so we have \(e = 4\), \(d = 6\), and \(q = 5\), in the terminology of [GPPS99]. We check [GPPS99, Ex. 2.1–2.9]. Ex. 2.1 does not apply because the classes \(C_5\) and \(C_8\) are empty by [KL90, Table 3.5.C], Ex. 2.2 yields exactly the stabilizer \(G_2\) of a \(2\)-dimensional subspace, Ex. 2.4 (b) does not apply because \(d/\gcd(d,e)\) is odd, and all other cases are excluded by the fact that \(r = 3e+1\).
The group \(G_2\) has the structure \(2.(PSp(2,5) \times PSp(4,5))\), which is a central product of \(Sp(2,5) \cong 2.A_5\) and \(Sp(4,5) = 2.S_4(5)\) (see [KL90, Prop. 4.1.3]). The character table of \(G_2\) can be derived from that of the direct product of \(2.A_5\) and \(2.S_4(5)\), by factoring out the diagonal central subgroup of order two.
gap> t:= CharacterTable( "S6(5)" );; gap> s1:= CharacterTable( "2.A5" );; gap> s2:= CharacterTable( "2.S4(5)" );; gap> dp:= s1 * s2; CharacterTable( "2.A5x2.S4(5)" ) gap> c:= Difference( ClassPositionsOfCentre( dp ), Union( > GetFusionMap( s1, dp ), GetFusionMap( s2, dp ) ) ); [ 62 ] gap> s:= dp / c; CharacterTable( "2.A5x2.S4(5)/[ 1, 62 ]" )
Now we compute \(σ(S,s)\).
gap> SigmaFromMaxes( t, "78A", [ s ], [ 1 ] ); 9/217
We show that \(S = S_8(3) = PSp(8,3)\) satisfies the following.
For \(s \in S\) irreducible of order \(41\), \(𝕄(S,s)\) consists of one group \(M\) of the type \(S_4(9).2 = PSp(4,9).2\).
\(σ(S,s) = 1/546\).
The preimage of \(s\) in the matrix group \(2.S_8(3) = Sp(8,3)\) can be chosen of order \(82\), and the preimage of \(M\) is \(2.S_4(9).2 = Sp(4,9).2\).
By [Ber00], the only maximal subgroups of \(S\) that contain irreducible elements of order \((3^4+1)/2 = 41\) are of extension field type, and by [KL90, Prop. 4.3.10], these groups have the structure \(S_4(9).2\) and there is exactly one class of these groups.
The group \(U = S_4(9)\) has three nontrivial outer automorphisms, the character table of the subgroup \(U.2\) in question has the identifier "S4(9).2_1", which follows from the fact that the extensions of \(U\) by the other two outer automorphisms do not admit a class fusion into \(S\).
gap> t:= CharacterTable( "S8(3)" );; gap> pi:= List( [ "S4(9).2_1", "S4(9).2_2", "S4(9).2_3" ], > name -> PossiblePermutationCharacters( > CharacterTable( name ), t ) );; gap> List( pi, Length ); [ 1, 0, 0 ]
Now statement (a) follows from the fact that \((1_{U.2})^S(s) = 1\) holds.
gap> spos:= Position( OrdersClassRepresentatives( t ), 41 );; gap> pi[1][1][ spos ]; 1
Now we compute \(σ(S,s)\) in order to show statement (b).
gap> Maximum( ApproxP( pi[1], spos ) ); 1/546
Statement (c) is clear from the description of extension field type subgroups in [KL90].
We show that \(S = U_4(4) = SU(4,4)\) satisfies the following.
For \(s \in S\) of the type \(1 \perp 3\) (i. e., \(s\) decomposes the natural \(4\)-dimensional module for \(SU(4,4)\) into an orthogonal sum of two irreducible modules of the dimensions \(1\) and \(3\), respectively) and of order \(4^3+1 = 65\), \(𝕄(S,s)\) consists of one group of the type \(G_1 = 5 \times U_3(4) = GU(3,4)\).
\(σ(S,s) = 209/3\,264\).
By [MSW94], the only maximal subgroups of \(S\) that contain \(s\) are one class of stabilizers \(H \cong 5 \times U_3(4)\) of this decomposition, and clearly there is only one such group containing \(s\).
Note that \(H\) has index \(3\,264\) in \(S\), since \(S\) has two orbits on the \(1\)-dimensional subspaces, of lengths \(1\,105\) and \(3\,264\), respectively, and elements of order \(13 = 65/5\) lie in the stabilizers of points in the latter orbit.
gap> g:= SU(4,4);; gap> orbs:= OrbitsDomain( g, NormedRowVectors( GF(16)^4 ), OnLines );; gap> orblen:= List( orbs, Length ); [ 1105, 3264 ] gap> List( orblen, x -> x mod 13 ); [ 0, 1 ]
We compute the permutation character \(1_{G_1}^S\); there is exactly one combinatorially possible permutation character of degree \(3\,264\) (cf. [BP98]).
gap> t:= CharacterTable( "U4(4)" );; gap> pi:= PermChars( t, rec( torso:= [ orblen[2] ] ) );; gap> Length( pi ); 1
Now we compute \(σ(S,s)\).
gap> spos:= Position( OrdersClassRepresentatives( t ), 65 );; gap> Maximum( ApproxP( pi, spos ) ); 209/3264
We show that \(S = U_6(2) = PSU(6,2)\) satisfies the following.
For \(s \in S\) of order \(11\), \(𝕄(S,s)\) consists of one group of the type \(U_5(2) = SU(5,2)\) and three groups of the type \(M_{22}\).
\(σ(S,s) = 5/21\).
The preimage of \(s\) in the matrix group \(SU(6,2) = 3.U_6(2)\) can be chosen of order \(33\), and the preimages of the groups in \(𝕄(S,s)\) have the structures \(3 \times U_5(2) \cong GU(5,2)\) and \(3.M_{22}\), respectively.
With \(s\) as in (a), the automorphic extensions \(S.2\), \(S.3\) of \(S\) satisfy \(σ^{\prime}(S.2,s) = 5/96\) and \(σ^{\prime}(S.3,s) = 59/224\).
According to the list of maximal subgroups of \(S\) in [CCN+85, p. 115], \(s\) is contained exactly in maximal subgroups of the types \(U_5(2)\) (one class) and \(M_{22}\) (three classes).
The permutation character of the action on the cosets of \(U_5(2)\) type subgroups is uniquely determined by the character tables. We get three possibilities for the permutation character on the cosets of \(M_{22}\) type subgroups; they correspond to the three classes of such subgroups, because each of these classes contains elements in exactly one of the conjugacy classes 4C, 4D, and 4E of elements in \(S\), and these classes are fused under the outer automorphism of \(S\) of order three.
gap> t:= CharacterTable( "U6(2)" );; gap> s1:= CharacterTable( "U5(2)" );; gap> pi1:= PossiblePermutationCharacters( s1, t );; gap> Length( pi1 ); 1 gap> s2:= CharacterTable( "M22" );; gap> pi2:= PossiblePermutationCharacters( s2, t ); [ Character( CharacterTable( "U6(2)" ), [ 20736, 0, 384, 0, 0, 0, 54, 0, 0, 0, 0, 48, 0, 16, 6, 0, 0, 0, 0, 0, 0, 6, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( "U6(2)" ), [ 20736, 0, 384, 0, 0, 0, 54, 0, 0, 0, 48, 0, 0, 16, 6, 0, 0, 0, 0, 0, 0, 6, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( "U6(2)" ), [ 20736, 0, 384, 0, 0, 0, 54, 0, 0, 48, 0, 0, 0, 16, 6, 0, 0, 0, 0, 0, 0, 6, 0, 2, 0, 4, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ] gap> imgs:= Set( pi2, x -> Position( x, 48 ) ); [ 10, 11, 12 ] gap> AtlasClassNames( t ){ imgs }; [ "4C", "4D", "4E" ] gap> GetFusionMap( t, CharacterTable( "U6(2).3" ) ){ imgs }; [ 10, 10, 10 ] gap> prim:= Concatenation( pi1, pi2 );;
Now statement (a) follows from the fact that the permutation characters have the value \(1\) on \(s\).
gap> spos:= Position( OrdersClassRepresentatives( t ), 11 );; gap> List( prim, x -> x[ spos ] ); [ 1, 1, 1, 1 ]
For statement (b), we compute \(σ(S,s)\).
gap> Maximum( ApproxP( prim, spos ) ); 5/21
Statement (c) follows from [CCN+85], plus the information that \(3.U_6(2)\) does not contain groups of the structure \(3 \times M_{22}\).
gap> PossibleClassFusions( > CharacterTable( "Cyclic", 3 ) * CharacterTable( "M22" ), > CharacterTable( "3.U6(2)" ) ); [ ]
For statement (d), we need that the relevant maximal subgroups of \(S.2\) are \(U_5(2).2\) and one subgroup \(M_{22}.2\), and that the relevant maximal subgroup of \(S.3\) is \(U_5(2) \times 3\), see [CCN+85, p. 115].
gap> SigmaFromMaxes( CharacterTable( "U6(2).2" ), "11AB", > [ CharacterTable( "U5(2).2" ), CharacterTable( "M22.2" ) ], > [ 1, 1 ], "outer" ); 5/96 gap> SigmaFromMaxes( CharacterTable( "U6(2).3" ), "11A", > [ CharacterTable( "U5(2)" ) * CharacterTable( "Cyclic", 3 ) ], > [ 1 ], "outer" ); 59/224
Before we start the computations using groups, we clean the workspace.
gap> CleanWorkspace();
For alternating groups of odd degree \(n = 2m+1\), we choose \(s\) to be an \(n\)-cycle. The interesting cases in [BGK08, Proposition 6.7] are \(5 \leq n \leq 23\).
In each case, we compute representatives of the maximal subgroups of \(A_n\), consider only those that contain an \(n\)-cycle, and then compute the permutation characters. Additionally, we show also the names that are used for the subgroups in the GAP Library of Transitive Groups, see [Hul05] and the documentation of this library in the GAP Reference Manual.
gap> PrimitivesInfoForOddDegreeAlternatingGroup:= function( n ) > local G, max, cycle, spos, prim, nonz; > > G:= AlternatingGroup( n ); > > # Compute representatives of the classes of maximal subgroups. > max:= MaximalSubgroupClassReps( G ); > > # Omit subgroups that cannot contain an `n'-cycle. > max:= Filtered( max, m -> IsTransitive( m, [ 1 .. n ] ) ); > > # Compute the permutation characters. > cycle:= []; > cycle[ n-1 ]:= 1; > spos:= PositionProperty( ConjugacyClasses( CharacterTable( G ) ), > c -> CycleStructurePerm( Representative( c ) ) = cycle ); > prim:= List( max, m -> TrivialCharacter( m )^G ); > nonz:= PositionsProperty( prim, x -> x[ spos ] <> 0 ); > > # Compute the subgroup names and the multiplicities. > return rec( spos := spos, > prim := prim{ nonz }, > grps := List( max{ nonz }, > m -> TransitiveGroup( n, > TransitiveIdentification( m ) ) ), > mult := List( prim{ nonz }, x -> x[ spos ] ) ); > end;;
The sets \(𝕄/~(s)\) and the values \(σ(A_n,s)\) are as follows. For each degree in question, the first list shows names for representatives of the conjugacy classes of maximal subgroups containing a fixed \(n\)-cycle, and the second list shows the number of conjugates in each class.
gap> for n in [ 5, 7 .. 23 ] do > prim:= PrimitivesInfoForOddDegreeAlternatingGroup( n ); > bound:= Maximum( ApproxP( prim.prim, prim.spos ) ); > Print( n, ": ", prim.grps, ", ", prim.mult, ", ", bound, "\n" ); > od; 5: [ D(5) = 5:2 ], [ 1 ], 1/3 7: [ L(7) = L(3,2), L(7) = L(3,2) ], [ 1, 1 ], 2/5 9: [ 1/2[S(3)^3]S(3), L(9):3=P|L(2,8) ], [ 1, 3 ], 9/35 11: [ M(11), M(11) ], [ 1, 1 ], 2/105 13: [ F_78(13)=13:6, L(13)=PSL(3,3), L(13)=PSL(3,3) ], [ 1, 2, 2 ], 4/ 1155 15: [ 1/2[S(3)^5]S(5), 1/2[S(5)^3]S(3), L(15)=A_8(15)=PSL(4,2), L(15)=A_8(15)=PSL(4,2) ], [ 1, 1, 1, 1 ], 29/273 17: [ L(17):4=PYL(2,16), L(17):4=PYL(2,16) ], [ 1, 1 ], 2/135135 19: [ F_171(19)=19:9 ], [ 1 ], 1/6098892800 21: [ t21n150, t21n161, t21n91 ], [ 1, 1, 2 ], 29/285 23: [ M(23), M(23) ], [ 1, 1 ], 2/130945815
In the above output, a subgroup printed as 1/2[S(\(n_1\))^\(n_2\)]S(\(n_2\)), 1/2[S(\(n_1\))^\(n_2\)]S(\(n_2\)), where \(n = n_1 n_2\) holds, denotes the intersection of \(A_n\) with the wreath product \(S_{n_1} \wr S_{n_2} \leq S_n\). (Note that the Atlas denotes the subgroup 1/2[S(3)^3]S(3) of \(A_9\) as \(3^3:S_4\).) The groups printed as P|L(2,8) and PYL(2,16) denote \(PΓL(2,8)\) and \(PΓL(2,16)\), respectively. And the three subgroups of \(A_{21}\) have the structures \((S_3 \wr S_7) \cap A_{21}\), \((S_7 \wr S_3) \cap A_{21}\), and \(PGL(3,4)\), respectively.
Note that \(A_9\) contains two conjugacy classes of maximal subgroups of the type \(PΓL(2,8) \cong L_2(8):3\), and that each \(9\)-cycle in \(A_9\) is contained in exactly three conjugate subgroups of this type. For \(n \in \{ 13, 15, 17 \}\), \(A_n\) contains two conjugacy classes of isomorphic maximal subgroups of linear type, and each \(n\)-cycle is contained in subgroups from each class. Finally, \(A_{21}\) contains only one class of maximal subgroups of linear type.
For the two groups \(A_5\) and \(A_7\), the values computed above are not sufficient. See Section 11.5-2 and 11.5-4 for a further treatment.
The above computations look like a brute-force approach, but note that the computation of the maximal subgroups of alternating and symmetric groups in GAP uses the classification of these subgroups, and also the conjugacy classes of elements in alternating and symmetric groups can be computed cheaply.
Alternative (character-theoretic) computations for \(n \in \{ 5, 7, 9, 11, 13 \}\) were shown in Section 11.4-3. (A hand calculation for the case \(n = 19\) can be found in [BW75].)
We show that \(S = A_5\) satisfies the following.
\(σ(S) = 1/3\), and this value is attained exactly for \(σ(S,s)\) with \(s\) of order \(5\).
For \(s \in S\) of order \(5\), \(𝕄(S,s)\) consists of one group of the type \(D_{10}\).
\(P(S) = 1/3\), and this value is attained exactly for \(P(S,s)\) with \(s\) of order \(5\).
Each element in \(S\) together with one of \((1,2)(3,4)\), \((1,3)(2,4)\), \((1,4)(2,3)\) generates a proper subgroup of \(S\).
Both the spread and the uniform spread of \(S\) is exactly two (see [BW75]), with \(s\) of order \(5\).
Statement (a) follows from inspection of the primitive permutation characters, cf. Section 11.4-3.
gap> t:= CharacterTable( "A5" );; gap> ProbGenInfoSimple( t ); [ "A5", 1/3, 2, [ "5A" ], [ 1 ] ]
Statement (b) can be read off from the primitive permutation characters, and the fact that the unique class of maximal subgroups that contain elements of order \(5\) consists of groups of the structure \(D_{10}\), see [CCN+85, p. 2].
gap> OrdersClassRepresentatives( t ); [ 1, 2, 3, 5, 5 ] gap> PrimitivePermutationCharacters( t ); [ Character( CharacterTable( "A5" ), [ 5, 1, 2, 0, 0 ] ), Character( CharacterTable( "A5" ), [ 6, 2, 0, 1, 1 ] ), Character( CharacterTable( "A5" ), [ 10, 2, 1, 0, 0 ] ) ]
For statement (c), we compute that for all nonidentity elements \(s \in S\) and involutions \(g \in S\), \(P(g,s) \geq 1/3\) holds, with equality if and only if \(s\) has order \(5\). We actually compute, for class representatives \(s\), the proportion of involutions \(g\) such that \(\langle g, s \rangle ≠ S\) holds.
gap> g:= AlternatingGroup( 5 );; gap> inv:= g.1^2 * g.2; (1,4)(2,5) gap> cclreps:= List( ConjugacyClasses( g ), Representative );; gap> SortParallel( List( cclreps, Order ), cclreps ); gap> List( cclreps, Order ); [ 1, 2, 3, 5, 5 ] gap> Size( ConjugacyClass( g, inv ) ); 15 gap> prop:= List( cclreps, > r -> RatioOfNongenerationTransPermGroup( g, inv, r ) ); [ 1, 1, 3/5, 1/3, 1/3 ] gap> Minimum( prop ); 1/3
Statement (d) follows by explicit computations.
gap> triple:= [ (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) ];; gap> CommonGeneratorWithGivenElements( g, cclreps, triple ); fail
As for statement (e), we know from (a) that the uniform spread of \(S\) is at least two, and from (d) that the spread is less than three.
We show that \(S = A_6\) satisfies the following.
\(σ(S) = 2/3\), and this value is attained exactly for \(σ(S,s)\) with \(s\) of order \(5\).
For \(s\) of order \(5\), \(𝕄(S,s)\) consists of two nonconjugate groups of the type \(A_5\).
\(P(S) = 5/9\), and this value is attained exactly for \(P(S,s)\) with \(s\) of order \(5\).
Each element in \(S\) together with one of \((1,2)(3,4)\), \((1,3)(2,4)\), \((1,4)(2,3)\) generates a proper subgroup of \(S\).
Both the spread and the uniform spread of \(S\) is exactly two (see [BW75]), with \(s\) of order \(4\).
For \(x\), \(y \in S_6^{\times}\), there is \(s \in S_6\) such that \(S \subseteq \langle x, s \rangle \cap \langle y, s \rangle\). It is not possible to find \(s \in S\) with this property, or \(s\) in a prescribed conjugacy class of \(S_6\).
\(σ( PGL(2,9) ) = 1/6\) and \(σ( M_{10} ) = 1/9\), with \(s\) of order \(10\) and \(8\), respectively.
(Note that in this example, the optimal choice of \(s\) for \(P(S)\) cannot be used to obtain the result on the exact spread.)
Statement (a) follows from inspection of the primitive permutation characters, cf. Section 11.4-3.
gap> t:= CharacterTable( "A6" );; gap> ProbGenInfoSimple( t ); [ "A6", 2/3, 1, [ "5A" ], [ 2 ] ]
Statement (b) can be read off from the permutation characters, and the fact that the two classes of maximal subgroups that contain elements of order \(5\) consist of groups of the structure \(A_5\), see [CCN+85, p. 4].
gap> OrdersClassRepresentatives( t ); [ 1, 2, 3, 3, 4, 5, 5 ] gap> prim:= PrimitivePermutationCharacters( t ); [ Character( CharacterTable( "A6" ), [ 6, 2, 3, 0, 0, 1, 1 ] ), Character( CharacterTable( "A6" ), [ 6, 2, 0, 3, 0, 1, 1 ] ), Character( CharacterTable( "A6" ), [ 10, 2, 1, 1, 2, 0, 0 ] ), Character( CharacterTable( "A6" ), [ 15, 3, 3, 0, 1, 0, 0 ] ), Character( CharacterTable( "A6" ), [ 15, 3, 0, 3, 1, 0, 0 ] ) ]
For statement (c), we first compute that for all nonidentity elements \(s \in S\) and involutions \(g \in S\), \(P(g,s) \geq 5/9\) holds, with equality if and only if \(s\) has order \(5\). We actually compute, for class representatives \(s\), the proportion of involutions \(g\) such that \(\langle g, s \rangle ≠ S\) holds.
gap> S:= AlternatingGroup( 6 );; gap> inv:= (S.1*S.2)^2; (1,3)(2,5) gap> cclreps:= List( ConjugacyClasses( S ), Representative );; gap> SortParallel( List( cclreps, Order ), cclreps ); gap> List( cclreps, Order ); [ 1, 2, 3, 3, 4, 5, 5 ] gap> C:= ConjugacyClass( S, inv );; gap> Size( C ); 45 gap> prop:= List( cclreps, > r -> RatioOfNongenerationTransPermGroup( S, inv, r ) ); [ 1, 1, 1, 1, 29/45, 5/9, 5/9 ] gap> Minimum( prop ); 5/9
Now statement (c) follows from the fact that for \(g \in S\) of order larger than two, \(σ(S,g) \leq 1/2 < 5/9\) holds.
gap> ApproxP( prim, 6 ); [ 0, 2/3, 1/2, 1/2, 0, 1/3, 1/3 ]
Statement (d) follows by explicit computations.
gap> triple:= [ (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) ];; gap> CommonGeneratorWithGivenElements( S, cclreps, triple ); fail
An alternative triple to that in statement (d) is the one given in [BW75].
gap> triple:= [ (1,3)(2,4), (1,5)(2,6), (3,6)(4,5) ];; gap> CommonGeneratorWithGivenElements( S, cclreps, triple ); fail
Of course we can also construct such a triple, as follows.
gap> TripleWithProperty( [ [ inv ], C, C ], > l -> ForAll( S, elm -> > ForAny( l, x -> not IsGeneratorsOfTransPermGroup( S, [ elm, x ] ) ) ) ); [ (1,3)(2,5), (1,3)(2,6), (1,3)(2,4) ]
For statement (e), we use the random approach described in Section 11.3-3.
gap> s:= (1,2,3,4)(5,6);; gap> reps:= Filtered( cclreps, x -> Order( x ) > 1 );; gap> ResetGlobalRandomNumberGenerators(); gap> for pair in UnorderedTuples( reps, 2 ) do > if RandomCheckUniformSpread( S, pair, s, 40 ) <> true then > Print( "#E nongeneration!\n" ); > fi; > od;
We get no output, so a suitable element of order \(4\) works in all cases. Note that we cannot use an element of order \(5\), because it fixes a point in the natural permutation representation, and we may take \(x_1 = (1,2,3)\) and \(x_2 = (4,5,6)\). With this argument, only elements of order \(4\) and double \(3\)-cycles are possible choices for \(s\), and the latter are excluded by the fact that an outer automorphism maps the class of double \(s\)-cycles in \(A_6\) to the class of \(3\)-cycles. So no element in \(A_6\) of order different from \(4\) works.
Next we show statement (f). Already in \(A_6.2_1 = S_6\), elements \(s\) of order \(4\) do in general not work because they do not generate with transpositions.
gap> G:= SymmetricGroup( 6 );; gap> RatioOfNongenerationTransPermGroup( G, s, (1,2) ); 1
Also, choosing \(s\) from a prescribed conjugacy class of \(S_6\) (that is, also \(s\) outside \(A_6\) is allowed) with the property that \(A_6 \subseteq \langle x, s \rangle \cap \langle y, s \rangle\) is not possible. Note that only \(6\)-cycles are possible for \(s\) if \(x\) and \(y\) are commuting transpositions, and –applying the outer automorphism– no \(6\)-cycle works for two commuting fixed-point free involutions. (The group is small enough for a brute force test.)
gap> goods:= Filtered( Elements( G ), > s -> IsGeneratorsOfTransPermGroup( G, [ s, (1,2) ] ) and > IsGeneratorsOfTransPermGroup( G, [ s, (3,4) ] ) );; gap> Collected( List( goods, CycleStructurePerm ) ); [ [ [ ,,,, 1 ], 24 ] ] gap> goods:= Filtered( Elements( G ), > s -> IsGeneratorsOfTransPermGroup( G, [ s, (1,2)(3,4)(5,6) ] ) and > IsGeneratorsOfTransPermGroup( G, [ s, (1,3)(2,4)(5,6) ] ) );; gap> Collected( List( goods, CycleStructurePerm ) ); [ [ [ 1, 1 ], 24 ] ]
However, for each pair of nonidentity element \(x\), \(y \in S_6\), there is \(s \in S_6\) such that \(\langle x, s \rangle\) and \(\langle y, s \rangle\) both contain \(A_6\). (If \(s\) works for the pair \((x,y)\) then \(s^g\) works for \((x^g,y^g)\), so it is sufficient to consider only orbit representatives \((x,y)\) under the conjugation action of \(G\) on pairs. Thus we check conjugacy class representatives \(x\) and, for fixed \(x\), representatives of orbits of \(C_G(x)\) on the classes \(y^G\), i. e., representatives of \(C_G(y)\)-\(C_G(x)\)-double cosets in \(G\). Moreover, clearly we can restrict the checks to elements \(x\), \(y\) of prime order.)
gap> Sgens:= GeneratorsOfGroup( S );; gap> primord:= Filtered( List( ConjugacyClasses( G ), Representative ), > x -> IsPrimeInt( Order( x ) ) );; gap> for x in primord do > for y in primord do > for pair in DoubleCosetRepsAndSizes( G, Centralizer( G, y ), > Centralizer( G, x ) ) do > if not ForAny( G, s -> IsSubset( Group( x,s ), S ) and > IsSubset( Group( y^pair[1], s ), S ) ) then > Error( [ x, y ] ); > fi; > od; > od; > od;
In other words, the spread of \(S_6\) is \(2\) but the uniform spread of \(S_6\) is not \(2\) but only \(1\).
We cannot always find \(s \in A_6\) with the required property: If \(x\) is a transposition then any \(s\) with \(S \subseteq\langle x, s \rangle\) must be a \(5\)-cycle.
gap> filt:= Filtered( S, s -> IsSubset( Group( (1,2), s ), S ) );; gap> Collected( List( filt, Order ) ); [ [ 5, 48 ] ]
Moreover, clearly such \(s\) fixes one of the moved points of \(x\), so we may prescribe a transposition \(y ≠ x\) that commutes with \(x\), it satisfies \(S \nsubseteq\langle y, s \rangle\).
For the other two automorphic extensions \(A_6.2_2 = PGL(2,9)\) and \(A_6.2_3 = M_{10}\), we compute the character-theoretic bounds \(σ(A_6.2_2) = 1/6\) and \(σ(A_6.2_3) = 1/9\), which shows statement (g).
gap> ProbGenInfoSimple( CharacterTable( "A6.2_2" ) ); [ "A6.2_2", 1/6, 5, [ "10A" ], [ 1 ] ] gap> ProbGenInfoSimple( CharacterTable( "A6.2_3" ) ); [ "A6.2_3", 1/9, 8, [ "8C" ], [ 1 ] ]
Note that \(σ^{\prime}( PGL(2,9), s ) = 1/6\), with \(s\) of order \(5\), and \(σ^{\prime}( M_{10}, s ) = 0\) for any \(s \in A_6\) since \(M_{10}\) is a non-split extension of \(A_6\).
gap> t:= CharacterTable( "A6" );; gap> t2:= CharacterTable( "A6.2_2" );; gap> spos:= PositionsProperty( OrdersClassRepresentatives( t ), x -> x = 5 );; gap> ProbGenInfoAlmostSimple( t, t2, spos ); [ "A6.2_2", 1/6, [ "5A", "5B" ], [ 1, 1 ] ]
We show that \(S = A_7\) satisfies the following.
\(σ(S) = 2/5\), and this value is attained exactly for \(σ(S,s)\) with \(s\) of order \(7\).
For \(s\) of order \(7\), \(𝕄(S,s)\) consists of two nonconjugate subgroups of the type \(L_2(7)\).
\(P(S) = 2/5\), and this value is attained exactly for \(P(S,s)\) with \(s\) of order \(7\).
The uniform spread of \(S\) is exactly three, with \(s\) of order \(7\).
Statement (a) follows from inspection of the primitive permutation characters, cf. Section 11.4-3.
gap> t:= CharacterTable( "A7" );; gap> ProbGenInfoSimple( t ); [ "A7", 2/5, 2, [ "7A" ], [ 2 ] ]
Statement (b) can be read off from the permutation characters, and the fact that the two classes of maximal subgroups that contain elements of order \(7\) consist of groups of the structure \(L_2(7)\), see [CCN+85, p. 10].
gap> OrdersClassRepresentatives( t ); [ 1, 2, 3, 3, 4, 5, 6, 7, 7 ] gap> prim:= PrimitivePermutationCharacters( t ); [ Character( CharacterTable( "A7" ), [ 7, 3, 4, 1, 1, 2, 0, 0, 0 ] ), Character( CharacterTable( "A7" ), [ 15, 3, 0, 3, 1, 0, 0, 1, 1 ] ), Character( CharacterTable( "A7" ), [ 15, 3, 0, 3, 1, 0, 0, 1, 1 ] ), Character( CharacterTable( "A7" ), [ 21, 5, 6, 0, 1, 1, 2, 0, 0 ] ), Character( CharacterTable( "A7" ), [ 35, 7, 5, 2, 1, 0, 1, 0, 0 ] ) ]
For statement (c), we compute that for all nonidentity elements \(s \in S\) and involutions \(g \in S\), \(P(g,s) \geq 2/5\) holds, with equality if and only if \(s\) has order \(7\). We actually compute, for class representatives \(s\), the proportion of involutions \(g\) such that \(\langle g, s \rangle ≠ S\) holds.
gap> g:= AlternatingGroup( 7 );; gap> inv:= (g.1^3*g.2)^3; (2,6)(3,7) gap> ccl:= List( ConjugacyClasses( g ), Representative );; gap> SortParallel( List( ccl, Order ), ccl ); gap> List( ccl, Order ); [ 1, 2, 3, 3, 4, 5, 6, 7, 7 ] gap> Size( ConjugacyClass( g, inv ) ); 105 gap> prop:= List( ccl, r -> RatioOfNongenerationTransPermGroup( g, inv, r ) ); [ 1, 1, 1, 1, 89/105, 17/21, 19/35, 2/5, 2/5 ] gap> Minimum( prop ); 2/5
For statement (d), we use the random approach described in Section 11.3-3. By the character-theoretic bounds, it suffices to consider triples of elements in the classes 2A or 3B.
gap> OrdersClassRepresentatives( t ); [ 1, 2, 3, 3, 4, 5, 6, 7, 7 ] gap> spos:= Position( OrdersClassRepresentatives( t ), 7 );; gap> SizesCentralizers( t ); [ 2520, 24, 36, 9, 4, 5, 12, 7, 7 ] gap> ApproxP( prim, spos ); [ 0, 2/5, 0, 2/5, 2/15, 0, 0, 2/15, 2/15 ] gap> s:= (1,2,3,4,5,6,7);; gap> 3B:= (1,2,3)(4,5,6);; gap> C3B:= ConjugacyClass( g, 3B );; gap> Size( C3B ); 280 gap> ResetGlobalRandomNumberGenerators(); gap> for triple in UnorderedTuples( [ inv, 3B ], 3 ) do > if RandomCheckUniformSpread( g, triple, s, 80 ) <> true then > Print( "#E nongeneration!\n" ); > fi; > od;
We get no output, so the uniform spread of \(S\) is at least three.
Alternatively, we can use the lemma from Section 11.2-2; this approach is technically more involved but faster. We work with the diagonal product of the two degree \(15\) representations of \(S\), which is constructed from the information stored in the GAP Library of Tables of Marks.
gap> tom:= TableOfMarks( "A7" );; gap> a7:= UnderlyingGroup( tom );; gap> tommaxes:= MaximalSubgroupsTom( tom ); [ [ 39, 38, 37, 36, 35 ], [ 7, 15, 15, 21, 35 ] ] gap> index15:= List( tommaxes[1]{ [ 2, 3 ] }, > i -> RepresentativeTom( tom, i ) ); [ Group([ (1,3)(2,7), (1,5,7)(3,4,6) ]), Group([ (1,4)(2,3), (2,4,6)(3,5,7) ]) ] gap> deg15:= List( index15, s -> RightTransversal( a7, s ) );; gap> reps:= List( deg15, l -> Action( a7, l, OnRight ) ); [ Group([ (1,5,7)(2,9,10)(3,11,4)(6,12,8)(13,14,15), (1,8,15,5,12) (2,13,11,3,10)(4,14,9,7,6) ]), Group([ (1,2,3)(4,6,5)(7,8,9)(10,12,11)(13,15,14), (1,12,3,13,10) (2,9,15,4,11)(5,6,14,7,8) ]) ] gap> g:= DiagonalProductOfPermGroups( reps );; gap> ResetGlobalRandomNumberGenerators(); gap> repeat s:= Random( g ); > until Order( s ) = 7; gap> NrMovedPoints( s ); 28 gap> mpg:= MovedPoints( g );; gap> fixs:= Difference( mpg, MovedPoints( s ) );; gap> orb_s:= Orbit( g, fixs, OnSets );; gap> Length( orb_s ); 120 gap> SizesCentralizers( t ); [ 2520, 24, 36, 9, 4, 5, 12, 7, 7 ] gap> repeat 2a:= Random( g ); until Order( 2a ) = 2; gap> repeat 3b:= Random( g ); > until Order( 3b ) = 3 and Size( Centralizer( g, 3b ) ) = 9; gap> orb2a:= Orbit( g, Difference( mpg, MovedPoints( 2a ) ), OnSets );; gap> orb3b:= Orbit( g, Difference( mpg, MovedPoints( 3b ) ), OnSets );; gap> orb2aor3b:= Union( orb2a, orb3b );; gap> TripleWithProperty( [ [ orb2a[1], orb3b[1] ], orb2aor3b, orb2aor3b ], > l -> ForAll( orb_s, > f -> not IsEmpty( Intersection( Union( l ), f ) ) ) ); fail
It remains to show that for any choice of \(s \in S\), a quadruple of elements in \(S^{\times}\) exists such that \(s\) generates a proper subgroup of \(S\) together with at least one of these elements.
First we observe (without using GAP) that there is a pair of \(3\)-cycles whose fixed points cover the seven points of the natural permutation representation. This implies the statement for all elements \(s \in S\) that fix a point in this representation. So it remains to consider elements \(s\) of the orders six and seven.
For the order seven element, the above setup and the lemma from Section 11.2-2 can be used.
gap> QuadrupleWithProperty( [ [ orb2a[1] ], orb2a, orb2a, orb2a ], > l -> ForAll( orb_s, > f -> not IsEmpty( Intersection( Union( l ), f ) ) ) ); [ [ 2, 5, 12, 18, 19, 26 ], [ 7, 8, 9, 16, 21, 25 ], [ 1, 6, 10, 17, 20, 27 ], [ 13, 14, 15, 28, 29, 30 ] ]
For the order six element, we use the diagonal product of the primitive permutation representations of the degrees \(21\) and \(35\).
gap> has6A:= List( tommaxes[1]{ [ 4, 5 ] }, > i -> RepresentativeTom( tom, i ) ); [ Group([ (1,2)(3,7), (2,6,5,4)(3,7) ]), Group([ (2,3)(5,7), (1,2)(4,5,6,7), (2,3)(5,6) ]) ] gap> trans:= List( has6A, s -> RightTransversal( a7, s ) );; gap> reps:= List( trans, l -> Action( a7, l, OnRight ) ); [ Group([ (1,16,12)(2,17,13)(3,18,11)(4,19,14)(15,20,21), (1,4,7,9,10) (2,5,8,3,6)(11,12,15,14,13)(16,20,19,17,18) ]), Group([ (2,16,6)(3,17,7)(4,18,8)(5,19,9)(10,20,26)(11,21,27) (12,22,28)(13,23,29)(14,24,30)(15,25,31), (1,2,3,4,5) (6,10,13,15,9)(7,11,14,8,12)(16,20,23,25,19)(17,21,24,18,22) (26,32,35,31,28)(27,33,29,34,30) ]) ] gap> g:= DiagonalProductOfPermGroups( reps );; gap> repeat s:= Random( g ); > until Order( s ) = 6; gap> NrMovedPoints( s ); 53 gap> mpg:= MovedPoints( g );; gap> fixs:= Difference( mpg, MovedPoints( s ) );; gap> orb_s:= Orbit( g, fixs, OnSets );; gap> Length( orb_s ); 105 gap> repeat 3a:= Random( g ); > until Order( 3a ) = 3 and Size( Centralizer( g, 3a ) ) = 36; gap> orb3a:= Orbit( g, Difference( mpg, MovedPoints( 3a ) ), OnSets );; gap> Length( orb3a ); 35 gap> TripleWithProperty( [ [ orb3a[1] ], orb3a, orb3a ], > l -> ForAll( orb_s, > f -> not IsEmpty( Intersection( Union( l ), f ) ) ) ); [ [ 1, 4, 6, 12, 14, 15, 34, 37, 40, 43, 49 ], [ 1, 4, 6, 16, 19, 20, 27, 30, 33, 44, 49 ], [ 2, 3, 4, 5, 7, 9, 26, 47, 48, 50, 53 ] ]
So we have found not only a quadruple but even a triple of \(3\)-cycles that excludes candidates \(s\) of order six.
In the treatment of small dimensional linear groups \(S = SL(d,q)\), [BGK08] uses a Singer element \(s\) of order \((q^d-1)/(q-1)\). (So the order of the corresponding element in \(PSL(d,q) = (q^d-1)/[(q-1) \gcd(d,q-1)]\).) By [Ber00], \(𝕄(S,s)\) consists of extension field type subgroups, except in the cases \(d = 2\), \(q \in \{ 2, 5, 7, 9 \}\), and \((d,q) = (3,4)\). These subgroups have the structure \(GL(d/p,q^p):\alpha_q \cap S\), for prime divisors \(p\) of \(d\), where \(\alpha_q\) denotes the Frobenius automorphism that acts on matrices by raising each entry to the \(q\)-th power. (If \(q\) is a prime then we have \(GL(d/p,q^p):\alpha_q = ΓL(d/p,q^p)\).) Since \(s\) acts irreducibly, it is contained in at most one conjugate of each class of extension field type subgroups (cf. [BGK08, Lemma 2.12]).
First we write a GAP function RelativeSigmaL that takes a positive integer \(d\) and a basis \(B\) of the field extension of degree \(n\) over the field with \(q\) elements, and returns the group \(GL(d,q^n):\alpha_q\), as a subgroup of \(GL(dn,q)\).
gap> RelativeSigmaL:= function( d, B ) > local n, F, q, glgens, diag, pi, frob, i; > > n:= Length( B ); > F:= LeftActingDomain( UnderlyingLeftModule( B ) ); > q:= Size( F ); > > # Create the generating matrices inside the linear subgroup. > glgens:= List( GeneratorsOfGroup( SL( d, q^n ) ), > m -> BlownUpMat( B, m ) ); > > # Create the matrix of a diagonal part that maps to determinant 1. > diag:= IdentityMat( d*n, F ); > diag{ [ 1 .. n ] }{ [ 1 .. n ] }:= BlownUpMat( B, [ [ Z(q^n)^(q-1) ] ] ); > Add( glgens, diag ); > > # Create the matrix that realizes the Frobenius action, > # and adjust the determinant. > pi:= List( B, b -> Coefficients( B, b^q ) ); > frob:= NullMat( d*n, d*n, F ); > for i in [ 0 .. d-1 ] do > frob{ [ 1 .. n ] + i*n }{ [ 1 .. n ] + i*n }:= pi; > od; > diag:= IdentityMat( d*n, F ); > diag{ [ 1 .. n ] }{ [ 1 .. n ] }:= BlownUpMat( B, [ [ Z(q^n) ] ] ); > diag:= diag^LogFFE( Inverse( Determinant( frob ) ), Determinant( diag ) ); > > # Return the result. > return Group( Concatenation( glgens, [ diag * frob ] ) ); > end;;
The next function computes \(σ(SL(d,q),s)\), by computing the sum of \(μ(g,S/(GL(d/p,q^p):\alpha_q \cap S))\), for prime divisors \(p\) of \(d\), and taking the maximum over \(g \in S^{\times}\). The computations take place in a permutation representation of \(PSL(d,q)\).
gap> ApproxPForSL:= function( d, q ) > local G, epi, PG, primes, maxes, names, ccl; > > # Check whether this is an admissible case (see [Be00]). > if ( d = 2 and q in [ 2, 5, 7, 9 ] ) or ( d = 3 and q = 4 ) then > return fail; > fi; > > # Create the group SL(d,q), and the map to PSL(d,q). > G:= SL( d, q ); > epi:= ActionHomomorphism( G, NormedRowVectors( GF(q)^d ), OnLines ); > PG:= ImagesSource( epi ); > > # Create the subgroups corresponding to the prime divisors of `d'. > primes:= PrimeDivisors( d ); > maxes:= List( primes, p -> RelativeSigmaL( d/p, > Basis( AsField( GF(q), GF(q^p) ) ) ) ); > names:= List( primes, p -> Concatenation( "GL(", String( d/p ), ",", > String( q^p ), ").", String( p ) ) ); > if 2 < q then > names:= List( names, name -> Concatenation( name, " cap G" ) ); > fi; > > # Compute the conjugacy classes of prime order elements in the maxes. > # (In order to avoid computing all conjugacy classes of these subgroups, > # we work in Sylow subgroups.) > ccl:= List( List( maxes, x -> ImagesSet( epi, x ) ), > M -> ClassesOfPrimeOrder( M, PrimeDivisors( Size( M ) ), > TrivialSubgroup( M ) ) ); > > return [ names, UpperBoundFixedPointRatios( PG, ccl, true )[1] ]; > end;;
We apply this function to the cases that are interesting in [BGK08, Section 5.12].
gap> pairs:= [ [ 3, 2 ], [ 3, 3 ], [ 4, 2 ], [ 4, 3 ], [ 4, 4 ], > [ 6, 2 ], [ 6, 3 ], [ 6, 4 ], [ 6, 5 ], [ 8, 2 ], [ 10, 2 ] ];; gap> array:= [];; gap> for pair in pairs do > d:= pair[1]; q:= pair[2]; > approx:= ApproxPForSL( d, q ); > Add( array, [ Concatenation( "SL(", String(d), ",", String(q), ")" ), > (q^d-1)/(q-1), > approx[1], approx[2] ] ); > od; gap> oldsize:= SizeScreen();; gap> SizeScreen( [ 80 ] );; gap> PrintFormattedArray( array ); SL(3,2) 7 [ "GL(1,8).3" ] 1/4 SL(3,3) 13 [ "GL(1,27).3 cap G" ] 1/24 SL(4,2) 15 [ "GL(2,4).2" ] 3/14 SL(4,3) 40 [ "GL(2,9).2 cap G" ] 53/1053 SL(4,4) 85 [ "GL(2,16).2 cap G" ] 1/108 SL(6,2) 63 [ "GL(3,4).2", "GL(2,8).3" ] 365/55552 SL(6,3) 364 [ "GL(3,9).2 cap G", "GL(2,27).3 cap G" ] 22843/123845436 SL(6,4) 1365 [ "GL(3,16).2 cap G", "GL(2,64).3 cap G" ] 1/85932 SL(6,5) 3906 [ "GL(3,25).2 cap G", "GL(2,125).3 cap G" ] 1/484220 SL(8,2) 255 [ "GL(4,4).2" ] 1/7874 SL(10,2) 1023 [ "GL(5,4).2", "GL(2,32).5" ] 1/129794 gap> SizeScreen( oldsize );;
The only missing case for [BGK08] is \(S = L_3(4)\), for which \(𝕄(S,s)\) consists of three groups of the type \(L_3(2)\) (see [CCN+85, p. 23]). The group \(L_3(4)\) has been considered already in Section 11.4-3, where \(σ(S,s) = 1/5\) has been proved. Also the cases \(SL(3,3)\), \(SL(4,2) \cong A_8\), and \(SL(4,3)\) have been handled there.
An alternative character-theoretic proof for \(S = L_6(2)\) looks as follows. In this case, the subgroups in \(𝕄(S,s)\) have the types \(ΓL(3,4) \cong GL(3,4).2 \cong 3.L_3(4).3.2_2\) and \(ΓL(2,8) \cong GL(2,8).3 \cong (7 \times L_2(8)).3\).
gap> t:= CharacterTable( "L6(2)" );; gap> s1:= CharacterTable( "3.L3(4).3.2_2" );; gap> s2:= CharacterTable( "(7xL2(8)).3" );; gap> SigmaFromMaxes( t, "63A", [ s1, s2 ], [ 1, 1 ] ); 365/55552
For \(S = SL(d,q)\) with prime dimension \(d\), and \(s \in S\) a Singer cycle, we have \(𝕄(S,s) = \{ M \}\), where \(M = N_S(\langle s \rangle) \cong ΓL(1,q^d) \cap S\). So
\[ σ(g,s) = μ(g,S/M) = |g^S \cap M|/|g^S| < |M|/|g^S| \leq (q^d-1) \cdot d/|g^S| \]
holds for any \(g \in S \setminus Z(S)\), which implies \(σ( S, s ) < \max\{ (q^d-1) \cdot d/|g^S|; g \in S \setminus Z(S) \}\). The right hand side of this inequality is returned by the following function. In [BGK08, Lemma 3.8], the global upper bound \(1/q^d\) is derived for primes \(d \geq 5\).
gap> UpperBoundForSL:= function( d, q ) > local G, Msize, ccl; > > if not IsPrimeInt( d ) then > Error( "<d> must be a prime" ); > fi; > > G:= SL( d, q ); > Msize:= (q^d-1) * d; > ccl:= Filtered( ConjugacyClasses( G ), > c -> Msize mod Order( Representative( c ) ) = 0 > and Size( c ) <> 1 ); > > return Msize / Minimum( List( ccl, Size ) ); > end;;
The interesting values are \((d,q)\) with \(d \in \{ 5, 7, 11 \}\) and \(q \in \{ 2, 3, 4 \}\), and perhaps also \((d,q) \in \{ (3,2), (3,3) \}\). (Here we exclude \(SL(11,4)\) because writing down the conjugacy classes of this group would exceed the permitted memory.)
gap> NrConjugacyClasses( SL(11,4) ); 1397660 gap> pairs:= [ [ 3, 2 ], [ 3, 3 ], [ 5, 2 ], [ 5, 3 ], [ 5, 4 ], > [ 7, 2 ], [ 7, 3 ], [ 7, 4 ], > [ 11, 2 ], [ 11, 3 ] ];; gap> array:= [];; gap> for pair in pairs do > d:= pair[1]; q:= pair[2]; > approx:= UpperBoundForSL( d, q ); > Add( array, [ Concatenation( "SL(", String(d), ",", String(q), ")" ), > (q^d-1)/(q-1), > approx ] ); > od; gap> PrintFormattedArray( array ); SL(3,2) 7 7/8 SL(3,3) 13 3/4 SL(5,2) 31 31/64512 SL(5,3) 121 10/81 SL(5,4) 341 15/256 SL(7,2) 127 7/9142272 SL(7,3) 1093 14/729 SL(7,4) 5461 21/4096 SL(11,2) 2047 2047/34112245508649716682268134604800 SL(11,3) 88573 22/59049
The exact values are clearly better than the above bounds. We compute them for \(L_5(2)\) and \(L_7(2)\). In the latter case, the class fusion of the \(127:7\) type subgroup \(M\) is not uniquely determined by the character tables; here we use the additional information that the elements of order \(7\) in \(M\) have centralizer order \(49\) in \(L_7(2)\). (See Section 11.4-3 for the examples with \(d = 3\).)
gap> SigmaFromMaxes( CharacterTable( "L5(2)" ), "31A", > [ CharacterTable( "31:5" ) ], [ 1 ] ); 1/5376 gap> t:= CharacterTable( "L7(2)" );; gap> s:= CharacterTable( "P:Q", [ 127, 7 ] );; gap> pi:= PossiblePermutationCharacters( s, t );; gap> Length( pi ); 2 gap> ord7:= PositionsProperty( OrdersClassRepresentatives( t ), x -> x = 7 ); [ 38, 45, 76, 77, 83 ] gap> sizes:= SizesCentralizers( t ){ ord7 }; [ 141120, 141120, 3528, 3528, 49 ] gap> List( pi, x -> x[83] ); [ 42, 0 ] gap> spos:= Position( OrdersClassRepresentatives( t ), 127 );; gap> Maximum( ApproxP( pi{ [ 1 ] }, spos ) ); 1/4388290560
For the following values of \(d\) and \(q\), automorphic extensions \(G\) of \(L_d(q)\) had to be checked for [BGK08, Section 5.12].
\[ (d,q) \in \{ (3,4), (6,2), (6,3), (6,4), (6,5), (10,2) \} \]
The first case has been treated in Section 11.4-4. For the other cases, we compute \(σ^{\prime}(G,s)\) below.
In any case, the extension by a graph automorphism occurs, which can be described by mapping each matrix in \(SL(d,q)\) to its inverse transpose. If \(q > 2\), also extensions by diagonal automorphisms occur, which are induced by conjugation with elements in \(GL(d,q)\). If \(q\) is nonprime then also extensions by field automorphisms occur, which can be described by powering the matrix entries by roots of \(q\). Finally, products (of prime order) of these three kinds of automorphisms have to be considered.
We start with the extension \(G\) of \(S = SL(d,q)\) by a graph automorphism. \(G\) can be embedded into \(GL(2d,q)\) by representing the matrix \(A \in S\) as a block diagonal matrix with diagonal blocks equal to \(A\) and \(A^{-tr}\), and representing the graph automorphism by a permutation matrix that interchanges the two blocks. In order to construct the field extension type subgroups of \(G\), we have to choose the basis of the field extension in such a way that the subgroup is normalized by the permutation matrix; a sufficient condition is that the matrices of the \(𝔽_q\)-linear mappings induced by the basis elements are symmetric.
(We do not give a function that computes a basis with this property from the parameters \(d\) and \(q\). Instead, we only write down the bases that we will need.)
gap> SymmetricBasis:= function( q, n ) > local vectors, B, issymmetric; > > if q = 2 and n = 2 then > vectors:= [ Z(2)^0, Z(2^2) ]; > elif q = 2 and n = 3 then > vectors:= [ Z(2)^0, Z(2^3), Z(2^3)^5 ]; > elif q = 2 and n = 5 then > vectors:= [ Z(2)^0, Z(2^5), Z(2^5)^4, Z(2^5)^25, Z(2^5)^26 ]; > elif q = 3 and n = 2 then > vectors:= [ Z(3)^0, Z(3^2) ]; > elif q = 3 and n = 3 then > vectors:= [ Z(3)^0, Z(3^3)^2, Z(3^3)^7 ]; > elif q = 4 and n = 2 then > vectors:= [ Z(2)^0, Z(2^4)^3 ]; > elif q = 4 and n = 3 then > vectors:= [ Z(2)^0, Z(2^3), Z(2^3)^5 ]; > elif q = 5 and n = 2 then > vectors:= [ Z(5)^0, Z(5^2)^2 ]; > elif q = 5 and n = 3 then > vectors:= [ Z(5)^0, Z(5^3)^9, Z(5^3)^27 ]; > else > Error( "sorry, no basis for <q> and <n> stored" ); > fi; > > B:= Basis( AsField( GF(q), GF(q^n) ), vectors ); > > # Check that the basis really has the required property. > issymmetric:= M -> M = TransposedMat( M ); > if not ForAll( B, b -> issymmetric( BlownUpMat( B, [ [ b ] ] ) ) ) then > Error( "wrong basis!" ); > fi; > > # Return the result. > return B; > end;;
In later examples, we will need similar embeddings of matrices. Therefore, we provide a more general function EmbeddedMatrix that takes a field F, a matrix mat, and a function func, and returns a block diagonal matrix over F whose diagonal blocks are mat and func( mat ).
gap> BindGlobal( "EmbeddedMatrix", function( F, mat, func ) > local d, result; > > d:= Length( mat ); > result:= NullMat( 2*d, 2*d, F ); > result{ [ 1 .. d ] }{ [ 1 .. d ] }:= mat; > result{ [ d+1 .. 2*d ] }{ [ d+1 .. 2*d ] }:= func( mat ); > > return result; > end );
The following function is similar to ApproxPForSL, the differences are that the group \(G\) in question is not \(SL(d,q)\) but the extension of this group by a graph automorphism, and that \(σ^{\prime}(G,s)\) is computed not \(σ(G,s)\).
gap> ApproxPForOuterClassesInExtensionOfSLByGraphAut:= function( d, q ) > local embedG, swap, G, orb, epi, PG, Gprime, primes, maxes, ccl, names; > > # Check whether this is an admissible case (see [Be00], > # note that a graph automorphism exists only for `d > 2'). > if d = 2 or ( d = 3 and q = 4 ) then > return fail; > fi; > > # Provide a function that constructs a block diagonal matrix. > embedG:= mat -> EmbeddedMatrix( GF( q ), mat, > M -> TransposedMat( M^-1 ) ); > > # Create the matrix that exchanges the two blocks. > swap:= NullMat( 2*d, 2*d, GF(q) ); > swap{ [ 1 .. d ] }{ [ d+1 .. 2*d ] }:= IdentityMat( d, GF(q) ); > swap{ [ d+1 .. 2*d ] }{ [ 1 .. d ] }:= IdentityMat( d, GF(q) ); > > # Create the group SL(d,q).2, and the map to the projective group. > G:= ClosureGroupDefault( Group( List( GeneratorsOfGroup( SL( d, q ) ), > embedG ) ), > swap ); > orb:= Orbit( G, One( G )[1], OnLines ); > epi:= ActionHomomorphism( G, orb, OnLines ); > PG:= ImagesSource( epi ); > Gprime:= DerivedSubgroup( PG ); > > # Create the subgroups corresponding to the prime divisors of `d'. > primes:= PrimeDivisors( d ); > maxes:= List( primes, > p -> ClosureGroupDefault( Group( List( GeneratorsOfGroup( > RelativeSigmaL( d/p, SymmetricBasis( q, p ) ) ), > embedG ) ), > swap ) ); > > # Compute conjugacy classes of outer involutions in the maxes. > # (In order to avoid computing all conjugacy classes of these subgroups, > # we work in the Sylow $2$ subgroups.) > maxes:= List( maxes, M -> ImagesSet( epi, M ) ); > ccl:= List( maxes, M -> ClassesOfPrimeOrder( M, [ 2 ], Gprime ) ); > names:= List( primes, p -> Concatenation( "GL(", String( d/p ), ",", > String( q^p ), ").", String( p ) ) ); > > return [ names, UpperBoundFixedPointRatios( PG, ccl, true )[1] ]; > end;;
And these are the results for the groups we are interested in (and others).
gap> ApproxPForOuterClassesInExtensionOfSLByGraphAut( 4, 3 ); [ [ "GL(2,9).2" ], 17/117 ] gap> ApproxPForOuterClassesInExtensionOfSLByGraphAut( 4, 4 ); [ [ "GL(2,16).2" ], 73/1008 ] gap> ApproxPForOuterClassesInExtensionOfSLByGraphAut( 6, 2 ); [ [ "GL(3,4).2", "GL(2,8).3" ], 41/1984 ] gap> ApproxPForOuterClassesInExtensionOfSLByGraphAut( 6, 3 ); [ [ "GL(3,9).2", "GL(2,27).3" ], 541/352836 ] gap> ApproxPForOuterClassesInExtensionOfSLByGraphAut( 6, 4 ); [ [ "GL(3,16).2", "GL(2,64).3" ], 3265/12570624 ] gap> ApproxPForOuterClassesInExtensionOfSLByGraphAut( 6, 5 ); [ [ "GL(3,25).2", "GL(2,125).3" ], 13001/195250000 ] gap> ApproxPForOuterClassesInExtensionOfSLByGraphAut( 8, 2 ); [ [ "GL(4,4).2" ], 367/1007872 ] gap> ApproxPForOuterClassesInExtensionOfSLByGraphAut( 10, 2 ); [ [ "GL(5,4).2", "GL(2,32).5" ], 609281/476346056704 ]
Now we consider diagonal automorphisms. We modify the approach for \(SL(d,q)\) by constructing the field extension type subgroups of \(GL(d,q) \ldots\)
gap> RelativeGammaL:= function( d, B ) > local n, F, q, diag; > > n:= Length( B ); > F:= LeftActingDomain( UnderlyingLeftModule( B ) ); > q:= Size( F ); > diag:= IdentityMat( d * n, F ); > diag{[ 1 .. n ]}{[ 1 .. n ]}:= BlownUpMat( B, [ [ Z(q^n) ] ] ); > return ClosureGroup( RelativeSigmaL( d, B ), diag ); > end;;
\(\ldots\) and counting the elements of prime order outside the simple group.
gap> ApproxPForOuterClassesInGL:= function( d, q ) > local G, epi, PG, Gprime, primes, maxes, names; > > # Check whether this is an admissible case (see [Be00]). > if ( d = 2 and q in [ 2, 5, 7, 9 ] ) or ( d = 3 and q = 4 ) then > return fail; > fi; > > # Create the group GL(d,q), and the map to PGL(d,q). > G:= GL( d, q ); > epi:= ActionHomomorphism( G, NormedRowVectors( GF(q)^d ), OnLines ); > PG:= ImagesSource( epi ); > Gprime:= ImagesSet( epi, SL( d, q ) ); > > # Create the subgroups corresponding to the prime divisors of `d'. > primes:= PrimeDivisors( d ); > maxes:= List( primes, p -> RelativeGammaL( d/p, > Basis( AsField( GF(q), GF(q^p) ) ) ) ); > maxes:= List( maxes, M -> ImagesSet( epi, M ) ); > names:= List( primes, p -> Concatenation( "M(", String( d/p ), ",", > String( q^p ), ")" ) ); > > return [ names, > UpperBoundFixedPointRatios( PG, List( maxes, > M -> ClassesOfPrimeOrder( M, > PrimeDivisors( Index( PG, Gprime ) ), Gprime ) ), > true )[1] ]; > end;;
Here are the required results.
gap> ApproxPForOuterClassesInGL( 6, 3 ); [ [ "M(3,9)", "M(2,27)" ], 41/882090 ] gap> ApproxPForOuterClassesInGL( 4, 3 ); [ [ "M(2,9)" ], 0 ] gap> ApproxPForOuterClassesInGL( 6, 4 ); [ [ "M(3,16)", "M(2,64)" ], 1/87296 ] gap> ApproxPForOuterClassesInGL( 6, 5 ); [ [ "M(3,25)", "M(2,125)" ], 821563/756593750000 ]
(Note that the extension field type subgroup in \(PGL(4,3) = L_4(3).2_1\) is a non-split extension of its intersection with \(L_4(3)\), hence the zero value.)
Concerning extensions by Frobenius automorphisms, only the case \((d,q) = (6,4)\) is interesting in [BGK08]. In fact, we would not need to compute anything for the extension \(G\) of \(S = SL(6,4)\) by the Frobenius map that squares each matrix entry. This is because \(𝕄^{\prime}(G,s)\) consists of the normalizers of the two subgroups of the types \(SL(3,16)\) and \(SL(2,64)\), and the former maximal subgroup is a non-split extension of its intersection with \(S\), so only one maximal subgroup can contribute to \(σ^{\prime}(G,s)\), which is thus smaller than \(1/2\), by [BGK08, Prop. 2.6].
However, it is easy enough to compute the exact value of \(σ^{\prime}(G,s)\). We work with the projective action of \(S\) on its natural module, and compute the permutation induced by the Frobenius map as the Frobenius action on the normed row vectors.
gap> matgrp:= SL(6,4);; gap> dom:= NormedRowVectors( GF(4)^6 );; gap> Gprime:= Action( matgrp, dom, OnLines );; gap> pi:= PermList( List( dom, v -> Position( dom, List( v, x -> x^2 ) ) ) );; gap> G:= ClosureGroup( Gprime, pi );;
Then we compute the maximal subgroups, the classes of outer involutions, and the bound, similar to the situation with graph automorphisms.
gap> maxes:= List( [ 2, 3 ], p -> Normalizer( G, > Action( RelativeSigmaL( 6/p, > Basis( AsField( GF(4), GF(4^p) ) ) ), dom, OnLines ) ) );; gap> ccl:= List( maxes, M -> ClassesOfPrimeOrder( M, [ 2 ], Gprime ) );; gap> List( ccl, Length ); [ 0, 1 ] gap> UpperBoundFixedPointRatios( G, ccl, true ); [ 1/34467840, true ]
For \((d,q) = (6,4)\), we have to consider also the extension \(G\) of \(S = SL(6,4)\) by the product \(\alpha\) of the Frobenius map and the graph automorphism. We use the same approach as for the graph automorphism, i. e., we embed \(SL(6,4)\) into a \(12\)-dimensional group of \(6 \times 6\) block matrices, where the second block is the image of the first block under \(\alpha\), and describe \(\alpha\) by the transposition of the two blocks.
First we construct the projective actions of \(S\) and \(G\) on an orbit of \(1\)-spaces.
gap> embedFG:= function( F, mat ) > return EmbeddedMatrix( F, mat, > M -> List( TransposedMat( M^-1 ), > row -> List( row, x -> x^2 ) ) ); > end;; gap> d:= 6;; q:= 4;; gap> alpha:= NullMat( 2*d, 2*d, GF(q) );; gap> alpha{ [ 1 .. d ] }{ [ d+1 .. 2*d ] }:= IdentityMat( d, GF(q) );; gap> alpha{ [ d+1 .. 2*d ] }{ [ 1 .. d ] }:= IdentityMat( d, GF(q) );; gap> Gprime:= Group( List( GeneratorsOfGroup( SL(d,q) ), > mat -> embedFG( GF(q), mat ) ) );; gap> G:= ClosureGroupDefault( Gprime, alpha );; gap> orb:= Orbit( G, One( G )[1], OnLines );; gap> G:= Action( G, orb, OnLines );; gap> Gprime:= Action( Gprime, orb, OnLines );;
Next we construct the maximal subgroups, the classes of outer involutions, and the bound.
gap> maxes:= List( PrimeDivisors( d ), p -> Group( List( GeneratorsOfGroup( > RelativeSigmaL( d/p, Basis( AsField( GF(q), GF(q^p) ) ) ) ), > mat -> embedFG( GF(q), mat ) ) ) );; gap> maxes:= List( maxes, x -> Action( x, orb, OnLines ) );; gap> maxes:= List( maxes, x -> Normalizer( G, x ) );; gap> ccl:= List( maxes, M -> ClassesOfPrimeOrder( M, [ 2 ], Gprime ) );; gap> List( ccl, Length ); [ 0, 1 ] gap> UpperBoundFixedPointRatios( G, ccl, true ); [ 1/10792960, true ]
The only missing cases are the extensions of \(SL(6,3)\) and \(SL(6,5)\) by the involutory outer automorphism that acts as the product of a diagonal and a graph automorphism.
In the case \(S = SL(6,3)\), we can directly write down the extension \(G\).
gap> d:= 6;; q:= 3;; gap> diag:= IdentityMat( d, GF(q) );; gap> diag[1][1]:= Z(q);; gap> embedDG:= mat -> EmbeddedMatrix( GF(q), mat, > M -> TransposedMat( M^-1 )^diag );; gap> Gprime:= Group( List( GeneratorsOfGroup( SL(d,q) ), embedDG ) );; gap> alpha:= NullMat( 2*d, 2*d, GF(q) );; gap> alpha{ [ 1 .. d ] }{ [ d+1 .. 2*d ] }:= IdentityMat( d, GF(q) );; gap> alpha{ [ d+1 .. 2*d ] }{ [ 1 .. d ] }:= IdentityMat( d, GF(q) );; gap> G:= ClosureGroupDefault( Gprime, alpha );;
The maximal subgroups are constructed as the normalizers in \(G\) of the extension field type subgroups in \(S\). We work with a permutation representation of \(G\).
gap> maxes:= List( PrimeDivisors( d ), p -> Group( List( GeneratorsOfGroup( > RelativeSigmaL( d/p, Basis( AsField( GF(q), GF(q^p) ) ) ) ), > embedDG ) ) );; gap> orb:= Orbit( G, One( G )[1], OnLines );; gap> G:= Action( G, orb, OnLines );; gap> Gprime:= Action( Gprime, orb, OnLines );; gap> maxes:= List( maxes, M -> Normalizer( G, Action( M, orb, OnLines ) ) );; gap> ccl:= List( maxes, M -> ClassesOfPrimeOrder( M, [ 2 ], Gprime ) );; gap> List( ccl, Length ); [ 1, 1 ] gap> UpperBoundFixedPointRatios( G, ccl, true ); [ 25/352836, true ]
For \(S = SL(6,5)\), this approach does not work because we cannot realize the diagonal involution by an involutory matrix. Instead, we consider the extension of \(GL(6,5) \cong 2.(2 \times L_6(5)).2\) by the graph automorphism \(\alpha\), which can be embedded into \(GL(12,5)\).
gap> d:= 6;; q:= 5;; gap> embedG:= mat -> EmbeddedMatrix( GF(q), > mat, M -> TransposedMat( M^-1 ) );; gap> Gprime:= Group( List( GeneratorsOfGroup( SL(d,q) ), embedG ) );; gap> maxes:= List( PrimeDivisors( d ), p -> Group( List( GeneratorsOfGroup( > RelativeSigmaL( d/p, Basis( AsField( GF(q), GF(q^p) ) ) ) ), > embedG ) ) );; gap> diag:= IdentityMat( d, GF(q) );; gap> diag[1][1]:= Z(q);; gap> diag:= embedG( diag );; gap> alpha:= NullMat( 2*d, 2*d, GF(q) );; gap> alpha{ [ 1 .. d ] }{ [ d+1 .. 2*d ] }:= IdentityMat( d, GF(q) );; gap> alpha{ [ d+1 .. 2*d ] }{ [ 1 .. d ] }:= IdentityMat( d, GF(q) );; gap> G:= ClosureGroupDefault( Gprime, alpha * diag );;
Now we switch to the permutation action of this group on the \(1\)-dimensional subspaces, thus factoring out the cyclic normal subgroup of order four. In this action, the involutory diagonal automorphism is represented by an involution, and we can proceed as above.
gap> orb:= Orbit( G, One( G )[1], OnLines );; gap> Gprime:= Action( Gprime, orb, OnLines );; gap> G:= Action( G, orb, OnLines );; gap> maxes:= List( maxes, M -> Action( M, orb, OnLines ) );; gap> extmaxes:= List( maxes, M -> Normalizer( G, M ) );; gap> ccl:= List( extmaxes, M -> ClassesOfPrimeOrder( M, [ 2 ], Gprime ) );; gap> List( ccl, Length ); [ 2, 1 ] gap> UpperBoundFixedPointRatios( G, ccl, true ); [ 3863/6052750000, true ]
In the same way, we can recheck the values for the extensions of \(SL(6,5)\) by the diagonal or by the graph automorphism.
gap> diag:= Permutation( diag, orb, OnLines );; gap> G:= ClosureGroupDefault( Gprime, diag );; gap> extmaxes:= List( maxes, M -> Normalizer( G, M ) );; gap> ccl:= List( extmaxes, M -> ClassesOfPrimeOrder( M, [ 2 ], Gprime ) );; gap> List( ccl, Length ); [ 3, 1 ] gap> UpperBoundFixedPointRatios( G, ccl, true ); [ 821563/756593750000, true ] gap> alpha:= Permutation( alpha, orb, OnLines );; gap> G:= ClosureGroupDefault( Gprime, alpha );; gap> extmaxes:= List( maxes, M -> Normalizer( G, M ) );; gap> ccl:= List( extmaxes, M -> ClassesOfPrimeOrder( M, [ 2 ], Gprime ) );; gap> List( ccl, Length ); [ 2, 2 ] gap> UpperBoundFixedPointRatios( G, ccl, true ); [ 13001/195250000, true ]
gap> t2:= CharacterTable( "L6(2).2" );; gap> map:= InverseMap( GetFusionMap( t, t2 ) );; gap> torso:= List( Concatenation( prim ), pi -> CompositionMaps( pi, map ) );; gap> ext:= List( torso, x -> PermChars( t2, rec( torso:= x ) ) ); [ [ Character( CharacterTable( "L6(2).2" ), [ 55552, 0, 128, 256, 337, 112, 22, 0, 0, 16, 0, 16, 2, 17, 0, 0, 8, 2, 4, 28, 0, 0, 0, 4, 1, 0, 1, 0, 0, 4, 0, 2, 2, 2, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1120, 192, 32, 0, 0, 40, 13, 0, 4, 6, 0, 4, 4, 4, 0, 2, 8, 5, 0, 2, 0, 0, 0, 0, 1, 0, 1, 0 ] ) ], [ Character( CharacterTable( "L6(2).2" ), [ 1904640, 0, 0, 512, 960, 0, 120, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 73, 24, 3, 0, 0, 15, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 960, 960, 0, 0, 0, 0, 24, 0, 12, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 3, 0, 0, 0 ] ) ] ] gap> sigma:= ApproxP( Concatenation( ext ), > Position( OrdersClassRepresentatives( t2 ), 63 ) );; gap> Maximum( sigma{ Difference( PositionsProperty( > OrdersClassRepresentatives( t2 ), IsPrimeInt ), > ClassPositionsOfDerivedSubgroup( t2 ) ) } ); 41/1984 -->
We show that \(S = L_3(2) = SL(3,2)\) satisfies the following.
\(σ(S) = 1/4\), and this value is attained exactly for \(σ(S,s)\) with \(s\) of order \(7\).
For \(s\) of order \(7\), \(𝕄(S,s)\) consists of one group of the type \(7:3\).
\(P(S) = 1/4\), and this value is attained exactly for \(P(S,s)\) with \(s\) of order \(7\).
The uniform spread of \(S\) is at exactly three, with \(s\) of order \(7\), and the spread of \(S\) is exactly four. (This had been left open in [BW75].)
(Note that in this example, the spread and the uniform spread differ.)
Statement (a) follows from inspection of the primitive permutation characters, cf. Section 11.4-3.
gap> t:= CharacterTable( "L3(2)" );; gap> ProbGenInfoSimple( t ); [ "L3(2)", 1/4, 3, [ "7A" ], [ 1 ] ]
Statement (b) can be read off from the permutation characters, and the fact that the unique class of maximal subgroups that contain elements of order \(7\) consists of groups of the structure \(7:3\), see [CCN+85, p. 3].
gap> OrdersClassRepresentatives( t ); [ 1, 2, 3, 4, 7, 7 ] gap> PrimitivePermutationCharacters( t ); [ Character( CharacterTable( "L3(2)" ), [ 7, 3, 1, 1, 0, 0 ] ), Character( CharacterTable( "L3(2)" ), [ 7, 3, 1, 1, 0, 0 ] ), Character( CharacterTable( "L3(2)" ), [ 8, 0, 2, 0, 1, 1 ] ) ]
For the other statements, we will use the primitive permutation representations on \(7\) and \(8\) points of \(S\) (computed from the GAP Library of Tables of Marks), and their diagonal products of the degrees \(14\) and \(15\).
gap> tom:= TableOfMarks( "L3(2)" );; gap> g:= UnderlyingGroup( tom ); Group([ (2,4)(5,7), (1,2,3)(4,5,6) ]) gap> mx:= MaximalSubgroupsTom( tom ); [ [ 14, 13, 12 ], [ 7, 7, 8 ] ] gap> maxes:= List( mx[1], i -> RepresentativeTom( tom, i ) );; gap> tr:= List( maxes, s -> RightTransversal( g, s ) );; gap> acts:= List( tr, x -> Action( g, x, OnRight ) );; gap> g7:= acts[1]; Group([ (3,4)(6,7), (1,3,2)(4,6,5) ]) gap> g8:= acts[3]; Group([ (1,6)(2,5)(3,8)(4,7), (1,7,3)(2,5,8) ]) gap> g14:= DiagonalProductOfPermGroups( acts{ [ 1, 2 ] } ); Group([ (3,4)(6,7)(11,13)(12,14), (1,3,2)(4,6,5)(8,11,9)(10,12,13) ]) gap> g15:= DiagonalProductOfPermGroups( acts{ [ 2, 3 ] } ); Group([ (4,6)(5,7)(8,13)(9,12)(10,15)(11,14), (1,4,2)(3,5,6)(8,14,10) (9,12,15) ])
First we compute that for all nonidentity elements \(s \in S\) and order three elements \(g \in S\), \(P(g,s) \geq 1/4\) holds, with equality if and only if \(s\) has order \(7\); this implies statement (c). We actually compute, for class representatives \(s\), the proportion of order three elements \(g\) such that \(\langle g, s \rangle ≠ S\) holds.
gap> ccl:= List( ConjugacyClasses( g7 ), Representative );; gap> SortParallel( List( ccl, Order ), ccl ); gap> List( ccl, Order ); [ 1, 2, 3, 4, 7, 7 ] gap> Size( ConjugacyClass( g7, ccl[3] ) ); 56 gap> prop:= List( ccl, > r -> RatioOfNongenerationTransPermGroup( g7, ccl[3], r ) ); [ 1, 5/7, 19/28, 2/7, 1/4, 1/4 ] gap> Minimum( prop ); 1/4
Now we show that the uniform spread of \(S\) is less than four. In any of the primitive permutation representations of degree seven, we find three involutions whose sets of fixed points cover the seven points. The elements \(s\) of order different from \(7\) in \(S\) fix a point in this representation, so each such \(s\) generates a proper subgroup of \(S\) together with one of the three involutions.
gap> x:= g7.1; (3,4)(6,7) gap> fix:= Difference( MovedPoints( g7 ), MovedPoints( x ) ); [ 1, 2, 5 ] gap> orb:= Orbit( g7, fix, OnSets ); [ [ 1, 2, 5 ], [ 1, 3, 4 ], [ 2, 3, 6 ], [ 2, 4, 7 ], [ 1, 6, 7 ], [ 3, 5, 7 ], [ 4, 5, 6 ] ] gap> Union( orb{ [ 1, 2, 5 ] } ) = [ 1 .. 7 ]; true
So we still have to exclude elements \(s\) of order \(7\). In the primitive permutation representation of \(S\) on eight points, we find four elements of order three whose sets of fixed points cover the set of all points that are moved by \(S\), so with each element of order seven in \(S\), one of them generates an intransitive group.
gap> three:= g8.2; (1,7,3)(2,5,8) gap> fix:= Difference( MovedPoints( g8 ), MovedPoints( three ) ); [ 4, 6 ] gap> orb:= Orbit( g8, fix, OnSets );; gap> QuadrupleWithProperty( [ [ fix ], orb, orb, orb ], > list -> Union( list ) = [ 1 .. 8 ] ); [ [ 4, 6 ], [ 1, 7 ], [ 3, 8 ], [ 2, 5 ] ]
Together with statement (a), this proves that the uniform spread of \(S\) is exactly three, with \(s\) of order seven.
Each element of \(S\) fixes a point in the permutation representation on \(15\) points. So for proving that the spread of \(S\) is less than five, it is sufficient to find a quintuple of elements whose sets of fixed points cover all \(15\) points. (From the permutation characters it is clear that four of these elements must have order three, and the fifth must be an involution.)
gap> x:= g15.1; (4,6)(5,7)(8,13)(9,12)(10,15)(11,14) gap> fixx:= Difference( MovedPoints( g15 ), MovedPoints( x ) ); [ 1, 2, 3 ] gap> orbx:= Orbit( g15, fixx, OnSets ); [ [ 1, 2, 3 ], [ 1, 4, 5 ], [ 1, 6, 7 ], [ 2, 4, 6 ], [ 3, 4, 7 ], [ 3, 5, 6 ], [ 2, 5, 7 ] ] gap> y:= g15.2; (1,4,2)(3,5,6)(8,14,10)(9,12,15) gap> fixy:= Difference( MovedPoints( g15 ), MovedPoints( y ) ); [ 7, 11, 13 ] gap> orby:= Orbit( g15, fixy, OnSets );; gap> QuadrupleWithProperty( [ [ fixy ], orby, orby, orby ], > l -> Difference( [ 1 .. 15 ], Union( l ) ) in orbx ); [ [ 7, 11, 13 ], [ 5, 8, 14 ], [ 1, 10, 15 ], [ 3, 9, 12 ] ]
It remains to show that the spread of \(S\) is (at least) four. By the consideration of permutation characters, we know that we can find a suitable order seven element for all quadruples in question except perhaps quadruples of order three elements. We show that for each such case, we can choose \(s\) of order four. Since \(𝕄(S,s)\) consists of two subgroups of the type \(S_4\), we work with the representation on \(14\) points.)
First we compute \(s\) and the \(S\)-orbit of its fixed points, and the \(S\)-orbit of the fixed points of an element \(x\) of order three. Then we prove that for each quadruple of conjugates of \(x\), the union of their fixed points intersects the fixed points of at least one conjugate of \(s\) trivially.
gap> ResetGlobalRandomNumberGenerators(); gap> repeat s:= Random( g14 ); > until Order( s ) = 4; gap> s; (1,3)(2,6,7,5)(9,11,10,12)(13,14) gap> fixs:= Difference( MovedPoints( g14 ), MovedPoints( s ) ); [ 4, 8 ] gap> orbs:= Orbit( g14, fixs, OnSets );; gap> Length( orbs ); 21 gap> three:= g14.2; (1,3,2)(4,6,5)(8,11,9)(10,12,13) gap> fix:= Difference( MovedPoints( g14 ), MovedPoints( three ) ); [ 7, 14 ] gap> orb:= Orbit( g14, fix, OnSets );; gap> Length( orb ); 28 gap> QuadrupleWithProperty( [ [ fix ], orb, orb, orb ], > l -> ForAll( orbs, o -> not IsEmpty( Intersection( o, > Union( l ) ) ) ) ); fail
By the lemma from Section 11.2-2, we are done.
We show that \(S = M_{11}\) satisfies the following.
\(σ(S) = 1/3\), and this value is attained exactly for \(σ(S,s)\) with \(s\) of order \(11\).
For \(s\) of order \(11\), \(𝕄(S,s)\) consists of one group of the type \(L_2(11)\).
\(P(S) = 1/3\), and this value is attained exactly for \(P(S,s)\) with \(s\) of order \(11\).
Both the uniform spread and the spread of \(S\) is exactly three, with \(s\) of order \(11\).
Statement (a) follows from inspection of the primitive permutation characters, cf. Section 11.4-1.
gap> t:= CharacterTable( "M11" );; gap> ProbGenInfoSimple( t ); [ "M11", 1/3, 2, [ "11A" ], [ 1 ] ]
Statement (b) can be read off from the permutation characters, and the fact that the unique class of maximal subgroups that contain elements of order \(11\) consists of groups of the structure \(L_2(11)\), see [CCN+85, p. 18].
gap> OrdersClassRepresentatives( t ); [ 1, 2, 3, 4, 5, 6, 8, 8, 11, 11 ] gap> PrimitivePermutationCharacters( t ); [ Character( CharacterTable( "M11" ), [ 11, 3, 2, 3, 1, 0, 1, 1, 0, 0 ] ), Character( CharacterTable( "M11" ), [ 12, 4, 3, 0, 2, 1, 0, 0, 1, 1 ] ), Character( CharacterTable( "M11" ), [ 55, 7, 1, 3, 0, 1, 1, 1, 0, 0 ] ), Character( CharacterTable( "M11" ), [ 66, 10, 3, 2, 1, 1, 0, 0, 0, 0 ] ), Character( CharacterTable( "M11" ), [ 165, 13, 3, 1, 0, 1, 1, 1, 0, 0 ] ) ] gap> Maxes( t ); [ "A6.2_3", "L2(11)", "3^2:Q8.2", "A5.2", "2.S4" ]
For the other statements, we will use the primitive permutation representations of \(S\) on \(11\) and \(12\) points (which are fetched from the Atlas of Group Representations [WWT+]), and their diagonal product.
gap> gens11:= OneAtlasGeneratingSet( "M11", NrMovedPoints, 11 ); rec( charactername := "1a+10a", constituents := [ 1, 2 ], contents := "core", generators := [ (2,10)(4,11)(5,7)(8,9), (1,4,3,8)(2,5,6,9) ], groupname := "M11", id := "", identifier := [ "M11", [ "M11G1-p11B0.m1", "M11G1-p11B0.m2" ], 1, 11 ], isPrimitive := true, maxnr := 1, p := 11, rankAction := 2, repname := "M11G1-p11B0", repnr := 1, size := 7920, stabilizer := "A6.2_3", standardization := 1, transitivity := 4, type := "perm" ) gap> g11:= GroupWithGenerators( gens11.generators );; gap> gens12:= OneAtlasGeneratingSet( "M11", NrMovedPoints, 12 );; gap> g12:= GroupWithGenerators( gens12.generators );; gap> g23:= DiagonalProductOfPermGroups( [ g11, g12 ] ); Group([ (2,10)(4,11)(5,7)(8,9)(12,17)(13,20)(16,18)(19,21), (1,4,3,8) (2,5,6,9)(12,17,18,15)(13,19)(14,20)(16,22,23,21) ])
First we compute that for all nonidentity elements \(s \in S\) and involutions \(g \in S\), \(P(g,s) \geq 1/3\) holds, with equality if and only if \(s\) has order \(11\); this implies statement (c). We actually compute, for class representatives \(s\), the proportion of involutions \(g\) such that \(\langle g, s \rangle ≠ S\) holds.
gap> inv:= g11.1; (2,10)(4,11)(5,7)(8,9) gap> ccl:= List( ConjugacyClasses( g11 ), Representative );; gap> SortParallel( List( ccl, Order ), ccl ); gap> List( ccl, Order ); [ 1, 2, 3, 4, 5, 6, 8, 8, 11, 11 ] gap> Size( ConjugacyClass( g11, inv ) ); 165 gap> prop:= List( ccl, > r -> RatioOfNongenerationTransPermGroup( g11, inv, r ) ); [ 1, 1, 1, 149/165, 25/33, 31/55, 23/55, 23/55, 1/3, 1/3 ] gap> Minimum( prop ); 1/3
For the first part of statement (d), we have to deal only with the case of triples of involutions.
The \(11\)-cycle \(s\) is contained in exactly one maximal subgroup of \(S\), of index \(12\). By Corollary 1 in Section 11.2-2, it is enough to show that in the primitive degree \(12\) representation of \(S\), the fixed points of no triple \((x_1, x_2, x_3)\) of involutions in \(S\) can cover all twelve points; equivalenly (considering complements), we show that there is no triple such that the intersection of the sets of moved points is empty.
gap> inv:= g12.1; (1,6)(2,9)(5,7)(8,10) gap> moved:= MovedPoints( inv ); [ 1, 2, 5, 6, 7, 8, 9, 10 ] gap> orb12:= Orbit( g12, moved, OnSets );; gap> Length( orb12 ); 165 gap> TripleWithProperty( [ orb12{[1]}, orb12, orb12 ], > list -> IsEmpty( Intersection( list ) ) ); fail
This implies that the uniform spread of \(S\) is at least three.
Now we show that there is a quadruple consisting of one element of order three and three involutions whose fixed points cover all points in the degree \(23\) representation constructed above; since the permutation character of this representation is strictly positive, this implies that \(S\) does not have spread four, by Corollary 2 in Section 11.2-2, and we have proved statement (d).
gap> inv:= g23.1; (2,10)(4,11)(5,7)(8,9)(12,17)(13,20)(16,18)(19,21) gap> moved:= MovedPoints( inv ); [ 2, 4, 5, 7, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 21 ] gap> orb23:= Orbit( g23, moved, OnSets );; gap> three:= ( g23.1*g23.2^2 )^2; (2,6,10)(4,8,7)(5,9,11)(12,17,23)(15,18,16)(19,21,22) gap> movedthree:= MovedPoints( three );; gap> QuadrupleWithProperty( [ [ movedthree ], orb23, orb23, orb23 ], > list -> IsEmpty( Intersection( list ) ) ); [ [ 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 16, 17, 18, 19, 21, 22, 23 ], [ 1, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 16, 17, 18, 20, 21 ], [ 1, 2, 3, 4, 5, 6, 7, 11, 12, 13, 14, 15, 18, 19, 20, 23 ], [ 1, 2, 3, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 20, 22, 23 ] ]
We show that \(S = M_{12}\) satisfies the following.
\(σ(S) = 1/3\), and this value is attained exactly for \(σ(S,s)\) with \(s\) of order \(10\).
For \(s \in S\) of order \(10\), \(𝕄(S,s)\) consists of two nonconjugate subgroups of the type \(A_6.2^2\), and one group of the type \(2 \times S_5\).
\(P(S) = 31/99\), and this value is attained exactly for \(P(S,s)\) with \(s\) of order \(10\).
The uniform spread of \(S\) is at least three, with \(s\) of order \(10\).
\(σ^{\prime}(Aut(S), s) = 4/99\).
Statement (a) follows from inspection of the primitive permutation characters, cf. Section 11.4-1.
gap> t:= CharacterTable( "M12" );; gap> ProbGenInfoSimple( t ); [ "M12", 1/3, 2, [ "10A" ], [ 3 ] ]
Statement (b) can be read off from the permutation characters, and the fact that the only classes of maximal subgroups that contain elements of order \(10\) consist of groups of the structures \(A_6.2^2\) (two classes) and \(2 \times S_5\) (one class), see [CCN+85, p. 33].
gap> spos:= Position( OrdersClassRepresentatives( t ), 10 ); 13 gap> prim:= PrimitivePermutationCharacters( t );; gap> List( prim, x -> x{ [ 1, spos ] } ); [ [ 12, 0 ], [ 12, 0 ], [ 66, 1 ], [ 66, 1 ], [ 144, 0 ], [ 220, 0 ], [ 220, 0 ], [ 396, 1 ], [ 495, 0 ], [ 495, 0 ], [ 1320, 0 ] ] gap> Maxes( t ); [ "M11", "M12M2", "A6.2^2", "M12M4", "L2(11)", "3^2.2.S4", "M12M7", "2xS5", "M8.S4", "4^2:D12", "A4xS3" ]
For statement (c) (which implies statement (d)), we use the primitive permutation representation on \(12\) points.
gap> g:= MathieuGroup( 12 ); Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6), (1,12)(2,11) (3,6)(4,8)(5,9)(7,10) ])
First we show that for \(s\) of order \(10\), \(P(S,s) = 31/99\) holds.
gap> approx:= ApproxP( prim, spos ); [ 0, 3/11, 1/3, 1/11, 1/132, 13/99, 13/99, 13/396, 1/132, 1/33, 1/33, 1/33, 13/396, 0, 0 ] gap> 2B:= g.2^2; (3,11)(4,5)(6,10)(7,8) gap> Size( ConjugacyClass( g, 2B ) ); 495 gap> ResetGlobalRandomNumberGenerators(); gap> repeat s:= Random( g ); > until Order( s ) = 10; gap> prop:= RatioOfNongenerationTransPermGroup( g, 2B, s ); 31/99 gap> Filtered( approx, x -> x >= prop ); [ 1/3 ]
Next we show that for \(s\) of order different from \(10\), \(P(g,s)\) is larger than \(31/99\) for suitable \(g \in S^{\times}\). Except for \(s\) in the class 6A (which fixes no point in the degree \(12\) representation), it suffices to consider \(g\) in the class 2B (with four fixed points).
gap> x:= g.2^2; (3,11)(4,5)(6,10)(7,8) gap> ccl:= List( ConjugacyClasses( g ), Representative );; gap> SortParallel( List( ccl, Order ), ccl ); gap> prop:= List( ccl, r -> RatioOfNongenerationTransPermGroup( g, x, r ) );; gap> SortedList( prop ); [ 7/55, 31/99, 5/9, 5/9, 39/55, 383/495, 383/495, 43/55, 29/33, 1, 1, 1, 1, 1, 1 ] gap> bad:= Filtered( prop, x -> x < 31/99 ); [ 7/55 ] gap> pos:= Position( prop, bad[1] );; gap> [ Order( ccl[ pos ] ), NrMovedPoints( ccl[ pos ] ) ]; [ 6, 12 ]
In the remaining case, we choose \(g\) in the class 2A (which is fixed point free).
gap> x:= g.3; (1,12)(2,11)(3,6)(4,8)(5,9)(7,10) gap> s:= ccl[ pos ];; gap> prop:= RatioOfNongenerationTransPermGroup( g, x, s ); 17/33 gap> prop > 31/99; true
Statement (e) has been shown already in Section 11.4-2.
We show that \(S = O_7(3)\) satisfies the following.
\(σ(S) = 199/351\), and this value is attained exactly for \(σ(S,s)\) with \(s\) of order \(14\).
For \(s \in S\) of order \(14\), \(𝕄(S,s)\) consists of one group of the type \(2.U_4(3).2_2 = \Omega^-(6,3).2\) and two nonconjugate groups of the type \(S_9\).
\(P(S) = 155/351\), and this value is attained exactly for \(P(S,s)\) with \(s\) of order \(14\).
The uniform spread of \(S\) is at least three, with \(s\) of order \(14\).
\(σ^{\prime}(Aut(S), s) = 1/3\).
Currently GAP provides neither the table of marks of \(S\) nor all character tables of its maximal subgroups. First we compute those primitive permutation characters of \(S\) that have the degrees \(351\) (point stabilizer \(2.U_4(3).2_2\)), \(364\) (point stabilizer \(3^5:U_4(2).2\)), \(378\) (point stabilizer \(L_4(3).2_2\)), \(1\,080\) (point stabilizer \(G_2(3)\), two classes), \(1\,120\) (point stabilizer \(3^{3+3}:L_3(3)\)), \(3\,159\) (point stabilizer \(S_6(2)\), two classes), \(12\,636\) (point stabilizer \(S_9\), two classes), \(22\,113\) (point stabilizer \((2^2 \times U_4(2)).2\), which extends to \(D_8 \times U_4(2).2\) in \(O_7(3).2\)), and \(28\,431\) (point stabilizer \(2^6:A_7\)).
(So we ignore the primitive permutation characters of the degrees \(3\,640\), \(265\,356\), and \(331\,695\). Note that the orders of the corresponding subgroups are not divisible by \(7\).)
gap> t:= CharacterTable( "O7(3)" );; gap> someprim:= [];; gap> pi:= PossiblePermutationCharacters( > CharacterTable( "2.U4(3).2_2" ), t );; Length( pi ); 1 gap> Append( someprim, pi ); gap> pi:= PermChars( t, rec( torso:= [ 364 ] ) );; Length( pi ); 1 gap> Append( someprim, pi ); gap> pi:= PossiblePermutationCharacters( > CharacterTable( "L4(3).2_2" ), t );; Length( pi ); 1 gap> Append( someprim, pi ); gap> pi:= PossiblePermutationCharacters( CharacterTable( "G2(3)" ), t ); [ Character( CharacterTable( "O7(3)" ), [ 1080, 0, 0, 24, 108, 0, 0, 0, 27, 18, 9, 0, 12, 4, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 3, 6, 0, 3, 2, 2, 2, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 4, 0, 3, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( "O7(3)" ), [ 1080, 0, 0, 24, 108, 0, 0, 27, 0, 18, 9, 0, 12, 4, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 3, 0, 0, 6, 3, 2, 2, 2, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 4, 3, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0 ] ) ] gap> Append( someprim, pi ); gap> pi:= PermChars( t, rec( torso:= [ 1120 ] ) );; Length( pi ); 1 gap> Append( someprim, pi ); gap> pi:= PossiblePermutationCharacters( CharacterTable( "S6(2)" ), t ); [ Character( CharacterTable( "O7(3)" ), [ 3159, 567, 135, 39, 0, 81, 0, 0, 27, 27, 0, 15, 3, 3, 7, 4, 0, 27, 0, 0, 0, 0, 0, 9, 3, 0, 9, 0, 3, 9, 3, 0, 2, 1, 1, 0, 0, 0, 3, 0, 2, 0, 0, 0, 3, 0, 0, 3, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( "O7(3)" ), [ 3159, 567, 135, 39, 0, 81, 0, 27, 0, 27, 0, 15, 3, 3, 7, 4, 0, 27, 0, 0, 0, 0, 0, 9, 3, 0, 9, 3, 0, 3, 9, 0, 2, 1, 1, 0, 0, 3, 0, 0, 2, 0, 0, 0, 3, 0, 3, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0 ] ) ] gap> Append( someprim, pi ); gap> pi:= PossiblePermutationCharacters( CharacterTable( "S9" ), t ); [ Character( CharacterTable( "O7(3)" ), [ 12636, 1296, 216, 84, 0, 81, 0, 0, 108, 27, 0, 6, 0, 12, 10, 1, 0, 27, 0, 0, 0, 0, 0, 9, 3, 0, 9, 0, 12, 9, 3, 0, 1, 0, 2, 0, 0, 0, 3, 1, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1 ] ), Character( CharacterTable( "O7(3)" ), [ 12636, 1296, 216, 84, 0, 81, 0, 108, 0, 27, 0, 6, 0, 12, 10, 1, 0, 27, 0, 0, 0, 0, 0, 9, 3, 0, 9, 12, 0, 3, 9, 0, 1, 0, 2, 0, 0, 3, 0, 1, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1 ] ) ] gap> Append( someprim, pi ); gap> t2:= CharacterTable( "O7(3).2" );; gap> s2:= CharacterTable( "Dihedral", 8 ) * CharacterTable( "U4(2).2" ); CharacterTable( "Dihedral(8)xU4(2).2" ) gap> pi:= PossiblePermutationCharacters( s2, t2 );; Length( pi ); 1 gap> pi:= RestrictedClassFunctions( pi, t );; gap> Append( someprim, pi ); gap> pi:= PossiblePermutationCharacters( > CharacterTable( "2^6:A7" ), t );; Length( pi ); 1 gap> Append( someprim, pi ); gap> List( someprim, x -> x[1] ); [ 351, 364, 378, 1080, 1080, 1120, 3159, 3159, 12636, 12636, 22113, 28431 ]
Note that in the three cases where two possible permutation characters were found, there are in fact two classes of subgroups that induce different permutation characters. For the subgroups of the types \(G_2(3)\) and \(S_6(2)\), this is stated in [CCN+85, p. 109], and for the subgroups of the type \(S_9\), this follows from the fact that each \(S_9\) type subgroup in \(S\) contains elements in exactly one of the classes 3D or 3E, and these two classes are fused by the outer automorphism of \(S\).
gap> cl:= PositionsProperty( AtlasClassNames( t ), > x -> x in [ "3D", "3E" ] ); [ 8, 9 ] gap> List( Filtered( someprim, x -> x[1] = 12636 ), pi -> pi{ cl } ); [ [ 0, 108 ], [ 108, 0 ] ] gap> GetFusionMap( t, t2 ){ cl }; [ 8, 8 ]
Now we compute the lower bounds for \(σ( S, s^{\prime} )\) that are given by the sublist someprim of the primitive permutation characters.
gap> spos:= Position( OrdersClassRepresentatives( t ), 14 ); 52 gap> Maximum( ApproxP( someprim, spos ) ); 199/351
This shows that \(σ( S, s ) = 199/351\) holds. For statement (a), we have to show that choosing \(s^{\prime}\) from another class than 14A yields a larger value for \(σ( S, s^{\prime} )\).
gap> approx:= List( [ 1 .. NrConjugacyClasses( t ) ], > i -> Maximum( ApproxP( someprim, i ) ) );; gap> PositionsProperty( approx, x -> x <= 199/351 ); [ 52 ]
Statement (b) can be read off from the permutation characters.
gap> pos:= PositionsProperty( someprim, x -> x[ spos ] <> 0 ); [ 1, 9, 10 ] gap> List( someprim{ pos }, x -> x{ [ 1, spos ] } ); [ [ 351, 1 ], [ 12636, 1 ], [ 12636, 1 ] ]
For statement (c), we first compute \(P(g, s)\) for \(g\) in the class 2A, via explicit computations with the group. For dealing with this case, we first construct a faithful permutation representation of \(O_7(3)\) from the natural matrix representation of \(SO(7,3)\).
gap> so73:= SpecialOrthogonalGroup( 7, 3 );; gap> o73:= DerivedSubgroup( so73 );; gap> orbs:= OrbitsDomain( o73, Elements( GF(3)^7 ) );; gap> Set( orbs, Length ); [ 1, 702, 728, 756 ] gap> g:= Action( o73, First( orbs, x -> Length( x ) = 702 ) );; gap> Size( g ) = Size( t ); true
A 2A element \(g\) can be found as the \(7\)-th power of any element of order \(14\) in \(S\).
gap> ResetGlobalRandomNumberGenerators(); gap> repeat s:= Random( g ); > until Order( s ) = 14; gap> 2A:= s^7;; gap> bad:= RatioOfNongenerationTransPermGroup( g, 2A, s ); 155/351 gap> bad > 1/3; true gap> approx:= ApproxP( someprim, spos );; gap> PositionsProperty( approx, x -> x >= 1/3 ); [ 2 ]
This shows that \(P(g,s) = 155/351 > 1/3\). Since \(σ( g, s ) < 1/3\) for all nonidentity \(g\) not in the class 2A, we have \(P( S, s ) = 155/351\). For statement (c), it remains to show that \(P( S, s^{\prime} )\) is larger than \(155/351\) whenever \(s^{\prime}\) is not of order \(14\). First we compute \(P( g, s^{\prime} )\), for \(g\) in the class 2A.
gap> consider:= RepresentativesMaximallyCyclicSubgroups( t ); [ 18, 19, 25, 26, 27, 30, 31, 32, 34, 35, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 56, 57, 58 ] gap> Length( consider ); 28 gap> consider:= ClassesPerhapsCorrespondingToTableColumns( g, t, consider );; gap> Length( consider ); 31 gap> consider:= List( consider, Representative );; gap> SortParallel( List( consider, Order ), consider ); gap> app2A:= List( consider, c -> > RatioOfNongenerationTransPermGroup( g, 2A, c ) );; gap> SortedList( app2A ); [ 1/3, 1/3, 155/351, 191/351, 67/117, 23/39, 23/39, 85/117, 10/13, 10/13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] gap> test:= PositionsProperty( app2A, x -> x <= 155/351 );; gap> List( test, i -> Order( consider[i] ) ); [ 13, 13, 14 ]
We see that only for \(s^{\prime}\) in one of the two (algebraically conjugate) classes of element order \(13\), \(P( S, s^{\prime} )\) has a chance to be smaller than \(155/351\). This possibility is now excluded by counting elements in the class 3A that do not generate \(S\) together with \(s^{\prime}\) of order \(13\).
gap> C3A:= First( ConjugacyClasses( g ), > c -> Order( Representative( c ) ) = 3 and Size( c ) = 7280 );; gap> repeat ss:= Random( g ); > until Order( ss ) = 13; gap> bad:= RatioOfNongenerationTransPermGroup( g, Representative( C3A ), ss ); 17/35 gap> bad > 155/351; true
Now we show statement (d): For each triple \((x_1, x_2, x_3)\) of nonidentity elements in \(S\), there is an element \(s\) in the class 14A such that \(\langle x_i, s \rangle = S\) holds for \(1 \leq i \leq 3\). We can read off from the character-theoretic data that only those triples have to be checked for which at least two elements are contained in the class 2A, and the third element lies in one of the classes 2A, 2B, 3B.
gap> approx:= ApproxP( someprim, spos );; gap> max:= Maximum( approx{ [ 3 .. Length( approx ) ] } ); 59/351 gap> 155 + 2*59 < 351; true gap> third:= PositionsProperty( approx, x -> 2 * 155/351 + x >= 1 ); [ 2, 3, 6 ] gap> ClassNames( t ){ third }; [ "2a", "2b", "3b" ]
We can find elements in the classes 2B and 3B as powers of arbitrary elements of the orders \(20\) and \(15\), respectively.
gap> ord20:= PositionsProperty( OrdersClassRepresentatives( t ), > x -> x = 20 ); [ 58 ] gap> PowerMap( t, 10 ){ ord20 }; [ 3 ] gap> repeat x:= Random( g ); > until Order( x ) = 20; gap> 2B:= x^10;; gap> C2B:= ConjugacyClass( g, 2B );; gap> ord15:= PositionsProperty( OrdersClassRepresentatives( t ), > x -> x = 15 ); [ 53 ] gap> PowerMap( t, 10 ){ ord15 }; [ 6 ] gap> repeat x:= Random( g ); > until Order( x ) = 15; gap> 3B:= x^5;; gap> C3B:= ConjugacyClass( g, 3B );;
The existence of \(s\) can be shown with the random approach described in Section 11.3-3.
gap> repeat s:= Random( g ); > until Order( s ) = 14; gap> RandomCheckUniformSpread( g, [ 2A, 2A, 2A ], s, 50 ); true gap> RandomCheckUniformSpread( g, [ 2B, 2A, 2A ], s, 50 ); true gap> RandomCheckUniformSpread( g, [ 3B, 2A, 2A ], s, 50 ); true
Finally, we show statement (e). Let \(G = Aut(S) = S.2\). By [CCN+85, p. 109], \(𝕄^{\prime}(G,s)\) consists of the extension of the \(2.U_4(3).2_1\) type subgroup. We compute the extension of the permutation character.
gap> prim:= someprim{ [ 1 ] }; [ Character( CharacterTable( "O7(3)" ), [ 351, 127, 47, 15, 27, 45, 36, 0, 0, 9, 0, 15, 3, 3, 7, 6, 19, 19, 10, 11, 12, 8, 3, 5, 3, 6, 1, 0, 0, 3, 3, 0, 1, 1, 1, 6, 3, 0, 0, 2, 2, 0, 3, 0, 3, 3, 0, 0, 1, 0, 0, 1, 0, 4, 4, 1, 2, 0 ] ) ] gap> spos:= Position( AtlasClassNames( t ), "14A" );; gap> t2:= CharacterTable( "O7(3).2" );; gap> map:= InverseMap( GetFusionMap( t, t2 ) );; gap> torso:= List( prim, pi -> CompositionMaps( pi, map ) );; gap> ext:= List( torso, x -> PermChars( t2, rec( torso:= x ) ) ); [ [ Character( CharacterTable( "O7(3).2" ), [ 351, 127, 47, 15, 27, 45, 36, 0, 9, 0, 15, 3, 3, 7, 6, 19, 19, 10, 11, 12, 8, 3, 5, 3, 6, 1, 0, 3, 0, 1, 1, 1, 6, 3, 0, 2, 2, 0, 3, 0, 3, 3, 0, 1, 0, 0, 1, 0, 4, 1, 2, 0, 117, 37, 21, 45, 1, 13, 5, 1, 9, 9, 18, 15, 1, 7, 9, 6, 4, 0, 3, 0, 3, 3, 6, 2, 2, 9, 6, 1, 3, 1, 4, 1, 2, 1, 1, 0, 3, 1, 0, 0, 0, 0, 1, 1, 0, 0 ] ) ] ] gap> approx:= ApproxP( Concatenation( ext ), > Position( AtlasClassNames( t2 ), "14A" ) );; gap> Maximum( approx{ Difference( > PositionsProperty( OrdersClassRepresentatives( t2 ), IsPrimeInt ), > ClassPositionsOfDerivedSubgroup( t2 ) ) } ); 1/3
We show that \(S = O_8^+(2) = \Omega^+(8,2)\) satisfies the following.
\(σ(S) = 334/315\), and this value is attained exactly for \(σ(S,s)\) with \(s\) of order \(15\).
For \(s \in S\) of order \(15\), \(𝕄(S,s)\) consists of one group of the type \(S_6(2)\), two conjugate groups of the type \(2^6:A_8\), two conjugate groups of the type \(A_9\), and one group of each of the types \((3 \times U_4(2)):2 = (3 \times \Omega^-(6,2)):2\) and \((A_5 \times A_5):2^2 = (\Omega^-(4,2) \times \Omega^-(4,2)):2^2\).
\(P(S) = 29/42\), and this value is attained exactly for \(P(S,s)\) with \(s\) of order \(15\).
Let \(x, y \in S\) such that \(x, y, x y\) lie in the unique involution class of length \(1\,575\) of \(S\). (This is the class 2A.) Then each element in \(S\) together with one of \(x\), \(y\), \(x y\) generates a proper subgroup of \(S\).
Both the spread and the uniform spread of \(S\) is exactly two, with \(s\) of order \(15\).
For each choice of \(s \in S\), there is an extension \(S.2\) such that for any element \(g\) in the (outer) class 2F, \(\langle s, g \rangle\) does not contain \(S\).
For an element \(s\) of order \(15\) in \(S\), either \(S\) is the only maximal subgroup of \(S.2\) that contains \(s\), or the maximal subgroups of \(S.2\) that contain \(s\) are \(S\) and the extensions of the subgroups listed in statement (b); these groups have the structures \(S_6(2) \times 2\), \(2^6:S_8\) (twice), \(S_9\) (twice), \(S_3 \times U_4(2).2\), and \(S_5 \wr 2\).
For \(s \in S\) of order \(15\) and arbitrary \(g \in S.3 \setminus S\), we have \(\langle s, g \rangle = S.3\).
If \(x\), \(y\) are nonidentity elements in \(Aut(S)\) then there is an element \(s\) of order \(15\) in \(S\) such that \(S \subseteq \langle x, s \rangle \cap \langle y, s \rangle\).
Statement (a) follows from inspection of the primitive permutation characters, cf. Section 11.4-3.
gap> t:= CharacterTable( "O8+(2)" );; gap> ProbGenInfoSimple( t ); [ "O8+(2)", 334/315, 0, [ "15A", "15B", "15C" ], [ 7, 7, 7 ] ]
Statement (b) can be read off from the permutation characters, and the fact that the only classes of maximal subgroups that contain elements of order \(15\) consist of groups of the structures as claimed, see [CCN+85, p. 85].
gap> prim:= PrimitivePermutationCharacters( t );; gap> spos:= Position( OrdersClassRepresentatives( t ), 15 );; gap> List( Filtered( prim, x -> x[ spos ] <> 0 ), l -> l{ [ 1, spos ] } ); [ [ 120, 1 ], [ 135, 2 ], [ 960, 2 ], [ 1120, 1 ], [ 12096, 1 ] ]
For the remaining statements, we take a primitive permutation representation on \(120\) points, and assume that the permutation character is 1a+35a+84a. (See [CCN+85, p. 85], note that the three classes of maximal subgroups of index \(120\) in \(S\) are conjugate under triality.)
gap> matgroup:= DerivedSubgroup( GeneralOrthogonalGroup( 1, 8, 2 ) );; gap> points:= NormedRowVectors( GF(2)^8 );; gap> orbs:= OrbitsDomain( matgroup, points );; gap> List( orbs, Length ); [ 135, 120 ] gap> g:= Action( matgroup, orbs[2] );; gap> Size( g ); 174182400 gap> pi:= Sum( Irr( t ){ [ 1, 3, 7 ] } ); Character( CharacterTable( "O8+(2)" ), [ 120, 24, 32, 0, 0, 8, 36, 0, 0, 3, 6, 12, 4, 8, 0, 0, 0, 10, 0, 0, 12, 0, 0, 8, 0, 0, 3, 6, 0, 0, 2, 0, 0, 2, 1, 2, 2, 3, 0, 0, 2, 0, 0, 0, 0, 0, 3, 2, 0, 0, 1, 0, 0 ] )
In order to show statement (c), we first observe that for \(s\) in the class 15A and \(g\) not in one of the classes 2A, 2B, 3A, \(σ(g,s) < 1/3\) holds, and for the exceptional three classes, we have \(σ(g,s) > 1/2\).
gap> approx:= ApproxP( prim, spos );; gap> testpos:= PositionsProperty( approx, x -> x >= 1/3 ); [ 2, 3, 7 ] gap> AtlasClassNames( t ){ testpos }; [ "2A", "2B", "3A" ] gap> approx{ testpos }; [ 254/315, 334/315, 1093/1120 ] gap> ForAll( approx{ testpos }, x -> x > 1/2 ); true
Now we compute the values \(P(g,s)\), for \(s\) in the class 15A and \(g\) in one of the classes 2A, 2B, 3A.
By our choice of the character of the permutation representation we use, the class 15A is determined as the unique class of element order \(15\) with one fixed point. (Note that the three classes of element order \(15\) in \(S\) are conjugate under triality.) A 2A element can be found as the fourth power of any element of order \(8\) in \(S\), a 3A element can be found as the fifth power of a 15A element, and a 2B element can be found as the sixth power of an element of order \(12\), with \(32\) fixed points.
gap> ResetGlobalRandomNumberGenerators(); gap> repeat s:= Random( g ); > until Order( s ) = 15 and NrMovedPoints( g ) = 1 + NrMovedPoints( s ); gap> 3A:= s^5;; gap> repeat x:= Random( g ); until Order( x ) = 8; gap> 2A:= x^4;; gap> repeat x:= Random( g ); until Order( x ) = 12 and > NrMovedPoints( g ) = 32 + NrMovedPoints( x^6 ); gap> 2B:= x^6;; gap> prop15A:= List( [ 2A, 2B, 3A ], > x -> RatioOfNongenerationTransPermGroup( g, x, s ) ); [ 23/35, 29/42, 149/224 ] gap> Maximum( prop15A ); 29/42
This means that for \(s\) in the class 15A, we have \(P( S, s ) = 29/42\), and the same holds for all \(s\) of order \(15\) since the three classes of element order \(15\) are conjugate under triality. Now we show that for \(s\) of order different from \(15\), the value \(P(g,s)\) is larger than \(29/42\), for \(g\) in one of the classes 2A, 2B, 3A, or their images under triality. This implies statement (c).
gap> test:= List( [ 2A, 2B, 3A ], x -> ConjugacyClass( g, x ) );; gap> ccl:= ConjugacyClasses( g );; gap> consider:= Filtered( ccl, c -> Size( c ) in List( test, Size ) );; gap> Length( consider ); 7 gap> filt:= Filtered( ccl, c -> ForAll( consider, cc -> > RatioOfNongenerationTransPermGroup( g, Representative( cc ), > Representative( c ) ) <= 29/42 ) );; gap> Length( filt ); 3 gap> List( filt, c -> Order( Representative( c ) ) ); [ 15, 15, 15 ]
Now we show statement (d). First we observe that all those Klein four groups in \(S\) whose involutions lie in the class 2A are conjugate in \(S\). Note that this is the unique class of length \(1\,575\) in \(S\), and also the unique class whose elements have \(24\) fixed points in the degree \(120\) permutation representation.
For that, we use the character table of \(S\) to read off that \(S\) contains exactly \(14\,175\) such subgroups, and we use the group to compute one such subgroup and its normalizer of index \(14\,175\).
gap> SizesConjugacyClasses( t ); [ 1, 1575, 3780, 3780, 3780, 56700, 2240, 2240, 2240, 89600, 268800, 37800, 340200, 907200, 907200, 907200, 2721600, 580608, 580608, 580608, 100800, 100800, 100800, 604800, 604800, 604800, 806400, 806400, 806400, 806400, 2419200, 2419200, 2419200, 7257600, 24883200, 5443200, 5443200, 6451200, 6451200, 6451200, 8709120, 8709120, 8709120, 1209600, 1209600, 1209600, 4838400, 7257600, 7257600, 7257600, 11612160, 11612160, 11612160 ] gap> NrPolyhedralSubgroups( t, 2, 2, 2 ); rec( number := 14175, type := "V4" ) gap> repeat x:= Random( g ); > until Order( x ) mod 2 = 0 > and NrMovedPoints( x^( Order(x)/2 ) ) = 120 - 24; gap> x:= x^( Order(x)/2 );; gap> repeat y:= x^Random( g ); > until NrMovedPoints( x*y ) = 120 - 24; gap> v4:= SubgroupNC( g, [ x, y ] );; gap> n:= Normalizer( g, v4 );; gap> Index( g, n ); 14175
We verify that the triple has the required property.
gap> maxorder:= RepresentativesMaximallyCyclicSubgroups( t );; gap> maxorderreps:= List( ClassesPerhapsCorrespondingToTableColumns( g, t, > maxorder ), Representative );; gap> Length( maxorderreps ); 28 gap> CommonGeneratorWithGivenElements( g, maxorderreps, [ x, y, x*y ] ); fail
For the simple group \(S\), it remains to show statement (e). We want to show that for any choice of two nonidentity elements \(x\), \(y\) in \(S\), there is an element \(s\) in the class 15A such that \(\langle s, x \rangle = \langle s, y \rangle = S\) holds. Only \(x\), \(y\) in the classes given by the list testpos must be considered, by the estimates \(σ(g,s)\).
We replace the values \(σ(g,s)\) by the exact values \(P(g,s)\), for \(g\) in one of these three classes. Each of the three classes is determined by its element order and its number of fixed points.
gap> reps:= List( ccl, Representative );; gap> bading:= List( testpos, i -> Filtered( reps, > r -> Order( r ) = OrdersClassRepresentatives( t )[i] and > NrMovedPoints( r ) = 120 - pi[i] ) );; gap> List( bading, Length ); [ 1, 1, 1 ] gap> bading:= List( bading, x -> x[1] );;
For each pair \((C_1, C_2)\) of classes represented by this list, we have to show that for any choice of elements \(x \in C_1\), \(y \in C_2\) there is \(s\) in the class 15A such that \(\langle s, x \rangle = \langle s, y \rangle = S\) holds. This is done with the random approach that is described in Section 11.3-3.
gap> for pair in UnorderedTuples( bading, 2 ) do > test:= RandomCheckUniformSpread( g, pair, s, 80 ); > if test <> true then > Error( test ); > fi; > od;
We get no error message, so statement (e) holds.
Now we turn to the automorphic extensions of \(S\). First we compute a permutation representation of \(SO^+(8,2) \cong S.2\) and an element \(g\) in the class 2F, which is the unique conjugacy class of size \(120\) in \(S.2\).
gap> matgrp:= SO(1,8,2);; gap> g2:= Image( IsomorphismPermGroup( matgrp ) );; gap> IsTransitive( g2, MovedPoints( g2 ) ); true gap> repeat x:= Random( g2 ); until Order( x ) = 14; gap> 2F:= x^7;; gap> Size( ConjugacyClass( g2, 2F ) ); 120
Only for \(s\) in six conjugacy classes of \(S\), there is a nonzero probability to have \(S.2 = \langle g, s \rangle\).
gap> der:= DerivedSubgroup( g2 );; gap> cclreps:= List( ConjugacyClasses( der ), Representative );; gap> nongen:= List( cclreps, > x -> RatioOfNongenerationTransPermGroup( g2, 2F, x ) );; gap> goodpos:= PositionsProperty( nongen, x -> x < 1 );; gap> invariants:= List( goodpos, i -> [ Order( cclreps[i] ), > Size( Centralizer( g2, cclreps[i] ) ), nongen[i] ] );; gap> SortedList( invariants ); [ [ 10, 20, 1/3 ], [ 10, 20, 1/3 ], [ 12, 24, 2/5 ], [ 12, 24, 2/5 ], [ 15, 15, 0 ], [ 15, 15, 0 ] ]
\(S\) contains three classes of element order \(10\), which are conjugate in \(S.3\). For a fixed extension of the type \(S.2\), the element \(s\) can be chosen only in two of these three classes, which means that there is another group of the type \(S.2\) (more precisely, another subgroup of index three in \(S.S_3\)) in which this choice of \(s\) is not suitable –note that the general aim is to find \(s \in S\) uniformly for all automorphic extensions of \(S\). Analogous statements hold for the other possibilities for \(s\), so statement (f) follows.
Statement (g) follows from the list of maximal subgroups in [CCN+85, p. 85].
Statement (h) follows from the fact that \(S\) is the only maximal subgroup of \(S.3\) that contains elements of order \(15\), according to the list of maximal subgroups in [CCN+85, p. 85]. Alternatively, if we do not want to assume this information, we can use explicit computations, as follows. All we have to check is that any element in the classes 3F and 3G generates \(S.3\) together with a fixed element of order \(15\) in \(S\).
We compute a permutation representation of \(S.3\) as the derived subgroup of a subgroup of the type \(S.S_3\) inside the sporadic simple Fischer group \(Fi_{22}\); these subgroups lie in the fourth class of maximal subgroups of \(Fi_{22}\), see [CCN+85, p. 163]. An element in the class 3F of \(S.3\) can be found as a power of an order \(21\) element, and an element in the class 3G can be found as the fourth power of a 12P element.
gap> aut:= Group( AtlasGenerators( "Fi22", 1, 4 ).generators );; gap> Size( aut ) = 6 * Size( t ); true gap> g3:= DerivedSubgroup( aut );; gap> orbs:= OrbitsDomain( g3, MovedPoints( g3 ) );; gap> List( orbs, Length ); [ 3150, 360 ] gap> g3:= Action( g3, orbs[2] );; gap> repeat s:= Random( g3 ); until Order( s ) = 15; gap> repeat x:= Random( g3 ); until Order( x ) = 21; gap> 3F:= x^7;; gap> RatioOfNongenerationTransPermGroup( g3, 3F, s ); 0 gap> repeat x:= Random( g3 ); > until Order( x ) = 12 and Size( Centralizer( g3, x^4 ) ) = 648; gap> 3G:= x^4;; gap> RatioOfNongenerationTransPermGroup( g3, 3G, s ); 0
Finally, consider statement (i). It implies that [BGK08, Corollary 1.5] holds for \(\Omega^+(8,2)\), with \(s\) of order \(15\). Note that by part (f), \(s\) cannot be chosen in a prescribed conjugacy class of \(S\) that is independent of the elements \(x\), \(y\).
If \(x\) and \(y\) lie in \(S\) then statement (i) follows from part (e), and by part (g), the case that \(x\) or \(y\) lie in \(S.3 \setminus S\) is also not a problem. We now show that also \(x\) or \(y\) in \(S.2 \setminus S\) is not a problem. Here we have to deal with the cases that \(x\) and \(y\) lie in the same subgroup of index \(3\) in \(Aut(S)\) or in different such subgroups. Actually we show that for each index \(3\) subgroup \(H = S.2 < Aut(S)\), we can choose \(s\) from two of the three classes of element order \(15\) in \(S\) such that \(S\) is the only maximal subgroup of \(H\) that contains \(s\), and thus \(\langle x, s \rangle\) contains \(H\), for any choice of \(x \in H \setminus S\).
For that, we note that no novelty in \(S.2\) contains elements of order \(15\), so all maximal subgroups of \(S.2\) that contain such elements –besides \(S\)– have one of the indices \(120, 135, 960, 1120\), or \(12096\), and point stabilizers of the types \(S_6(2) \times 2\), \(2^6:S_8\), \(S_9\), \(S_3 \times U_4(2):2\), or \(S_5 \wr 2\). We compute the corresponding permutation characters.
gap> t2:= CharacterTable( "O8+(2).2" );; gap> s:= CharacterTable( "S6(2)" ) * CharacterTable( "Cyclic", 2 );; gap> pi:= PossiblePermutationCharacters( s, t2 );; gap> prim:= pi;; gap> pi:= PermChars( t2, rec( torso:= [ 135 ] ) );; gap> Append( prim, pi ); gap> pi:= PossiblePermutationCharacters( CharacterTable( "A9.2" ), t2 );; gap> Append( prim, pi ); gap> s:= CharacterTable( "Dihedral(6)" ) * CharacterTable( "U4(2).2" );; gap> pi:= PossiblePermutationCharacters( s, t2 );; gap> Append( prim, pi ); gap> s:= CharacterTableWreathSymmetric( CharacterTable( "S5" ), 2 );; gap> pi:= PossiblePermutationCharacters( s, t2 );; gap> Append( prim, pi ); gap> Length( prim ); 5 gap> ord15:= PositionsProperty( OrdersClassRepresentatives( t2 ), > x -> x = 15 ); [ 39, 40 ] gap> List( prim, pi -> pi{ ord15 } ); [ [ 1, 0 ], [ 2, 0 ], [ 2, 0 ], [ 1, 0 ], [ 1, 0 ] ] gap> List( ord15, i -> Maximum( ApproxP( prim, i ) ) ); [ 307/120, 0 ]
Here it is appropriate to clean the workspace again.
gap> CleanWorkspace();
We show that \(S = O_8^+(3)\) satisfies the following.
\(σ(S) = 863/1820\), and this value is attained exactly for \(σ(S,s)\) with \(s\) of order \(20\).
For \(s \in S\) of order \(20\), \(𝕄(S,s)\) consists of two nonconjugate groups of the type \(O_7(3) = \Omega(7,3)\), two conjugate subgroups of the type \(3^6:L_4(3)\), two nonconjugate subgroups of the type \((A_4 \times U_4(2)):2\), and one subgroup of each of the types \(2.U_4(3).(2^2)_{122}\) and \((A_6 \times A_6):2^2\).
\(P(S) = 194/455\), and this value is attained exactly for \(P(S,s)\) with \(s\) of order \(20\).
The uniform spread of \(S\) is at least three, with \(s\) of order \(20\).
The preimage of \(s\) in the matrix group \(2.S = \Omega^+(8,3)\) can be chosen of order \(40\), and then the maximal subgroups of \(2.S\) containing \(s\) have the structures \(2.O_7(3)\), \(3^6:2.L_4(3)\), \(4.U_4(3).2^2 = SU(4,3).2^2\), \(2.(A_4 \times U_4(2)).2 = 2.(PSp(2,3) \otimes PSp(4,3)).2\), and \(2.(A_6 \times A_6):2^2 = 2.(\Omega^-(4,3) \times \Omega^-(4,3)):2^2\), respectively.
For \(s \in S\) of order \(20\), we have \(P^{\prime}(S.2_1, s) \in \{ 83/567, 574/1215 \}\), \(P^{\prime}(S.2_2, s) \in \{ 0, 1 \}\) (depending on the choice of \(s\)), and \(σ^{\prime}(S.3, s) = 0\).
Furthermore, for any choice of \(s^{\prime} \in S\), we have \(σ^{\prime}(S.2_2, s^{\prime}) = 1\) for some group \(S.2_2\). However, if it is allowed to choose \(s\) from an \(Aut(S)\)-class of elements of order \(20\) (and not from a fixed \(S\)-class) then we can achieve \(σ(g,s) = 0\) for any given \(g \in S.2_2 \setminus S\).
The maximal subgroups of \(S.2_1\) that contain an element of order \(20\) are either \(S\) and the extensions of the subgroups listed in statement (b) or they are \(S\) and \(L_4(3).2^2\), \(3^6:L_4(3).2\) (twice), \(2.U_4(3).(2^2)_{122}.2\), and \((A_6 \times A_6):2^2.2\).
In the former case, the groups have the structures \(O_7(3):2\) (twice), \(3^6:(L_4(3) \times 2)\) (twice), \(S_4 \times U_4(2).2\) (twice), \(2.U_4(3).(2^2)_{122}.2\), and \((A_6 \times A_6):2^2 \times 2\).
Statement (a) follows from inspection of the primitive permutation characters.
gap> t:= CharacterTable( "O8+(3)" );; gap> ProbGenInfoSimple( t ); [ "O8+(3)", 863/1820, 2, [ "20A", "20B", "20C" ], [ 8, 8, 8 ] ]
Also statement (b) follows from the information provided by the character table of \(S\) (cf. [CCN+85, p. 140]).
gap> prim:= PrimitivePermutationCharacters( t );; gap> ord:= OrdersClassRepresentatives( t );; gap> spos:= Position( ord, 20 );; gap> filt:= PositionsProperty( prim, x -> x[ spos ] <> 0 ); [ 1, 2, 7, 15, 18, 19, 24 ] gap> Maxes( t ){ filt }; [ "O7(3)", "O8+(3)M2", "3^6:L4(3)", "2.U4(3).(2^2)_{122}", "(A4xU4(2)):2", "O8+(3)M19", "(A6xA6):2^2" ] gap> prim{ filt }{ [ 1, spos ] }; [ [ 1080, 1 ], [ 1080, 1 ], [ 1120, 2 ], [ 189540, 1 ], [ 7960680, 1 ], [ 7960680, 1 ], [ 9552816, 1 ] ]
For statement (c), we first show that \(P(S,s) = 194/455\) holds. Since this value is larger than \(1/3\), we have to inspect only those classes \(g^S\) for which \(σ(g,s) \geq 1/3\) holds,
gap> ord:= OrdersClassRepresentatives( t );; gap> ord20:= PositionsProperty( ord, x -> x = 20 );; gap> cand:= [];; gap> for i in ord20 do > approx:= ApproxP( prim, i ); > Add( cand, PositionsProperty( approx, x -> x >= 1/3 ) ); > od; gap> cand; [ [ 2, 6, 7, 10 ], [ 3, 6, 8, 11 ], [ 4, 6, 9, 12 ] ] gap> AtlasClassNames( t ){ cand[1] }; [ "2A", "3A", "3B", "3E" ]
The three possibilities form one orbit under the outer automorphism group of \(S\).
gap> t3:= CharacterTable( "O8+(3).3" );; gap> tfust3:= GetFusionMap( t, t3 );; gap> List( cand, x -> tfust3{ x } ); [ [ 2, 4, 5, 6 ], [ 2, 4, 5, 6 ], [ 2, 4, 5, 6 ] ]
By symmetry, we may consider only the first possibility, and assume that \(s\) is in the class 20A.
We work with a permutation representation of degree \(1\,080\), and assume that the permutation character is 1a+260a+819a. (Note that all permutation characters of \(S\) of degree \(1\,080\) are conjugate under \(Aut(S)\).)
gap> g:= Action( SO(1,8,3), NormedRowVectors( GF(3)^8 ), OnLines );; gap> Size( g ); 9904359628800 gap> g:= DerivedSubgroup( g );; Size( g ); 4952179814400 gap> orbs:= OrbitsDomain( g, MovedPoints( g ) );; gap> List( orbs, Length ); [ 1080, 1080, 1120 ] gap> g:= Action( g, orbs[1] );; gap> PositionProperty( Irr( t ), chi -> chi[1] = 819 ); 9 gap> permchar:= Sum( Irr( t ){ [ 1, 2, 9 ] } ); Character( CharacterTable( "O8+(3)" ), [ 1080, 128, 0, 0, 24, 108, 135, 0, 0, 108, 0, 0, 27, 27, 0, 0, 18, 9, 12, 16, 0, 0, 4, 15, 0, 0, 20, 0, 0, 12, 11, 0, 0, 20, 0, 0, 15, 0, 0, 12, 0, 0, 2, 0, 0, 3, 3, 0, 0, 6, 6, 0, 0, 3, 2, 2, 2, 18, 0, 0, 9, 0, 0, 0, 0, 0, 0, 3, 3, 0, 0, 3, 0, 0, 12, 0, 0, 3, 0, 0, 0, 0, 0, 4, 3, 3, 0, 0, 1, 0, 0, 4, 0, 0, 1, 1, 2, 0, 0, 0, 0, 0, 3, 0, 0, 2, 0, 0, 5, 0, 0, 1, 0, 0 ] )
Now we show that for \(s\) in the class 20A (which fixes one point), the proportion of nongenerating elements \(g\) in one of the classes 2A, 3A, 3B, 3E has the maximum \(194/455\), which is attained exactly for 3A. (We find a 2A element as a power of \(s\), a 3A element as a power of any element of order \(18\), a 3B and a 3E element as elements with \(135\) and \(108\) fixed points, respectively, which occur as powers of suitable elements of order \(15\).)
gap> permchar{ ord20 }; [ 1, 0, 0 ] gap> AtlasClassNames( t )[ PowerMap( t, 10 )[ ord20[1] ] ]; "2A" gap> ord18:= PositionsProperty( ord, x -> x = 18 );; gap> Set( AtlasClassNames( t ){ PowerMap( t, 6 ){ ord18 } } ); [ "3A" ] gap> ord15:= PositionsProperty( ord, x -> x = 15 );; gap> PowerMap( t, 5 ){ ord15 }; [ 7, 8, 9, 10, 11, 12 ] gap> AtlasClassNames( t ){ [ 7 .. 12 ] }; [ "3B", "3C", "3D", "3E", "3F", "3G" ] gap> permchar{ [ 7 .. 12 ] }; [ 135, 0, 0, 108, 0, 0 ] gap> mp:= NrMovedPoints( g );; gap> ResetGlobalRandomNumberGenerators(); gap> repeat 20A:= Random( g ); > until Order( 20A ) = 20 and mp - NrMovedPoints( 20A ) = 1; gap> 2A:= 20A^10;; gap> repeat x:= Random( g ); until Order( x ) = 18; gap> 3A:= x^6;; gap> repeat x:= Random( g ); > until Order( x ) = 15 and mp - NrMovedPoints( x^5 ) = 135; gap> 3B:= x^5;; gap> repeat x:= Random( g ); > until Order( x ) = 15 and mp - NrMovedPoints( x^5 ) = 108; gap> 3E:= x^5;; gap> nongen:= List( [ 2A, 3A, 3B, 3E ], > c -> RatioOfNongenerationTransPermGroup( g, c, 20A ) ); [ 3901/9477, 194/455, 451/1092, 451/1092 ] gap> Maximum( nongen ); 194/455
Next we compute the values \(P(g,s)\), for \(g\) is in the class 3A and certain elements \(s\). It is enough to consider representatives \(s\) of maximally cyclic subgroups in \(S\), but here we can do better, as follows. Since 3A is the unique class of length \(72\,800\), it is fixed under \(Aut(S)\), so it is enough to consider one element \(s\) from each \(Aut(S)\)-orbit on the classes of \(S\). We use the class fusion between the character tables of \(S\) and \(Aut(S)\) for computing orbit representatives.
gap> maxorder:= RepresentativesMaximallyCyclicSubgroups( t );; gap> Length( maxorder ); 57 gap> autt:= CharacterTable( "O8+(3).S4" );; gap> fus:= PossibleClassFusions( t, autt );; gap> orbreps:= Set( fus, map -> Set( ProjectionMap( map ) ) ); [ [ 1, 2, 5, 6, 7, 13, 17, 18, 19, 20, 23, 24, 27, 30, 31, 37, 43, 46, 50, 54, 55, 56, 57, 58, 64, 68, 72, 75, 78, 84, 85, 89, 95, 96, 97, 100, 106, 112 ] ] gap> totest:= Intersection( maxorder, orbreps[1] ); [ 43, 50, 54, 56, 57, 64, 68, 75, 78, 84, 85, 89, 95, 97, 100, 106, 112 ] gap> Length( totest ); 17 gap> AtlasClassNames( t ){ totest }; [ "6Q", "6X", "6B1", "8A", "8B", "9G", "9K", "12A", "12D", "12J", "12K", "12O", "13A", "14A", "15A", "18A", "20A" ]
This means that we have to test one element of each of the element orders \(13\), \(14\), \(15\), and \(18\) (note that we know already a bound for elements of order \(20\)), plus certain elements of the orders \(6\), \(8\), \(9\), and \(12\) which can be identified by their centralizer orders and (for elements of order \(6\) and \(8\)) perhaps the centralizer orders of some powers.
gap> elementstotest:= [];; gap> for elord in [ 13, 14, 15, 18 ] do > repeat s:= Random( g ); > until Order( s ) = elord; > Add( elementstotest, s ); > od;
The next elements to be tested are in the classes 6B1 (centralizer order \(162\)), in one of 9G–9J (centralizer order \(729\)), in one of 9K–9N (centralizer order \(81\)), in one of 12A–12C (centralizer order \(1\,728\)), in one of 12D–12I (centralizer order \(432\)), in 12J (centralizer order \(192\)), in one of 12K–12N (centralizer order \(108\)), and in one of 12O–12T (centralizer order \(72\)).
gap> ordcent:= [ [ 6, 162 ], [ 9, 729 ], [ 9, 81 ], [ 12, 1728 ], > [ 12, 432 ], [ 12, 192 ], [ 12, 108 ], [ 12, 72 ] ];; gap> cents:= SizesCentralizers( t );; gap> for pair in ordcent do > Print( pair, ": ", AtlasClassNames( t ){ > Filtered( [ 1 .. Length( ord ) ], > i -> ord[i] = pair[1] and cents[i] = pair[2] ) }, "\n" ); > repeat s:= Random( g ); > until Order( s ) = pair[1] and Size( Centralizer( g, s ) ) = pair[2]; > Add( elementstotest, s ); > od; [ 6, 162 ]: [ "6B1" ] [ 9, 729 ]: [ "9G", "9H", "9I", "9J" ] [ 9, 81 ]: [ "9K", "9L", "9M", "9N" ] [ 12, 1728 ]: [ "12A", "12B", "12C" ] [ 12, 432 ]: [ "12D", "12E", "12F", "12G", "12H", "12I" ] [ 12, 192 ]: [ "12J" ] [ 12, 108 ]: [ "12K", "12L", "12M", "12N" ] [ 12, 72 ]: [ "12O", "12P", "12Q", "12R", "12S", "12T" ]
The next elements to be tested are in one of the classes 6Q–6S (centralizer order \(648\)).
gap> AtlasClassNames( t ){ Filtered( [ 1 .. Length( ord ) ], > i -> cents[i] = 648 and cents[ PowerMap( t, 2 )[i] ] = 52488 > and cents[ PowerMap( t, 3 )[i] ] = 26127360 ) }; [ "6Q", "6R", "6S" ] gap> repeat s:= Random( g ); > until Order( s ) = 6 and Size( Centralizer( g, s ) ) = 648 > and Size( Centralizer( g, s^2 ) ) = 52488 > and Size( Centralizer( g, s^3 ) ) = 26127360; gap> Add( elementstotest, s );
The next elements to be tested are in the class 6X–6A1 (centralizer order \(648\)).
gap> AtlasClassNames( t ){ Filtered( [ 1 .. Length( ord ) ], > i -> cents[i] = 648 and cents[ PowerMap( t, 2 )[i] ] = 52488 > and cents[ PowerMap( t, 3 )[i] ] = 331776 ) }; [ "6X", "6Y", "6Z", "6A1" ] gap> repeat s:= Random( g ); > until Order( s ) = 6 and Size( Centralizer( g, s ) ) = 648 > and Size( Centralizer( g, s^2 ) ) = 52488 > and Size( Centralizer( g, s^3 ) ) = 331776; gap> Add( elementstotest, s );
Finally, we add elements from the classes 8A and 8B.
gap> AtlasClassNames( t ){ Filtered( [ 1 .. Length( ord ) ], > i -> ord[i] = 8 and cents[ PowerMap( t, 2 )[i] ] = 13824 ) }; [ "8A" ] gap> repeat s:= Random( g ); > until Order( s ) = 8 and Size( Centralizer( g, s^2 ) ) = 13824; gap> Add( elementstotest, s ); gap> AtlasClassNames( t ){ Filtered( [ 1 .. Length( ord ) ], > i -> ord[i] = 8 and cents[ PowerMap( t, 2 )[i] ] = 1536 ) }; [ "8B" ] gap> repeat s:= Random( g ); > until Order( s ) = 8 and Size( Centralizer( g, s^2 ) ) = 1536; gap> Add( elementstotest, s );
Now we compute the ratios. It turns out that from these candidates, only elements \(s\) of the orders \(14\) and \(15\) satisfy \(P(g,s) < 194/455\).
gap> nongen:= List( elementstotest, > s -> RatioOfNongenerationTransPermGroup( g, 3A, s ) );; gap> smaller:= PositionsProperty( nongen, x -> x < 194/455 ); [ 2, 3 ] gap> nongen{ smaller }; [ 127/325, 1453/3640 ]
So the only candidates for \(s\) that may be better than order \(20\) elements are elements of order \(14\) or \(15\). In order to exclude these two possibilities, we compute \(P(g,s)\) for \(s\) in the class 14A and \(g = s^7\) in the class 2A, and for \(s\) in the class 15A and \(g\) in the class 2A, which yields values that are larger than \(194/455\).
gap> repeat s:= Random( g ); > until Order( s ) = 14 and NrMovedPoints( s ) = 1078; gap> 2A:= s^7;; gap> nongen:= RatioOfNongenerationTransPermGroup( g, 2A, s ); 1573/3645 gap> nongen > 194/455; true gap> repeat s:= Random( g ); > until Order( s ) = 15 and NrMovedPoints( s ) = 1080 - 3; gap> nongen:= RatioOfNongenerationTransPermGroup( g, 2A, s ); 490/1053 gap> nongen > 194/455; true
For statement (d), we show that for each triple of elements in the union of the classes 2A, 3A, 3B, 3E there is an element in the class 20A that generates \(S\) together with each element of the triple.
gap> for tup in UnorderedTuples( [ 2A, 3A, 3B, 3E ], 3 ) do > cl:= ShallowCopy( tup ); > test:= RandomCheckUniformSpread( g, cl, 20A, 100 ); > if test <> true then > Error( test ); > fi; > od;
We get no error message, so statement (d) is true.
For statement (e), first we show that \(2.S = \Omega^+(8,3)\) contains elements of order \(40\) but \(S\) does not.
gap> der:= DerivedSubgroup( SO(1,8,3) );; gap> repeat x:= PseudoRandom( der ); until Order( x ) = 40; gap> 40 in ord; false
Thus elements of order \(40\) must arise as preimages of order \(20\) elements under the natural epimorphism from \(2.S\) to \(S\), which means that we may choose an order \(40\) preimage \(\hat{s}\) of \(s\). Then \(𝕄(2.S, \hat{s})\) consists of central extensions of the subgroups listed in statement (b). The perfect subgroups \(O_7(3)\), \(L_4(3)\), \(2.U_4(3)\), and \(U_4(2)\) of these groups must lift to their Schur double covers in \(2.S\) because otherwise the preimages would not contain elements of order \(40\).
Next we consider the preimage of the subgroup \(U = (A_4 \times U_4(2)).2\) of \(S\). We show that the preimages of the two direct factors \(A_4\) and \(U_4(2)\) in \(U^{\prime} = A_4 \times U_4(2)\) are Schur covers. For \(A_4\), this follows from the fact that the preimage of \(U^{\prime}\) must contain elements of order \(20\), and that \(U_4(2)\) does not contain elements of order \(10\).
gap> u42:= CharacterTable( "U4(2)" );; gap> Filtered( OrdersClassRepresentatives( u42 ), x -> x mod 5 = 0 ); [ 5 ]
In order to show that the \(U_4(2)\) type subgroup of \(U^{\prime}\) lifts to its double cover in \(2.S\), we note that the class 2B of \(U_4(2)\) lifts to a class of elements of order four in the double cover \(2.U_4(2)\), and that the corresponding class of elements in \(U\) is \(S\)-conjugate to the class of involutions in the direct factor \(A_4\) (which is the unique class of length three in \(U\)).
gap> u:= CharacterTable( Maxes( t )[18] ); CharacterTable( "(A4xU4(2)):2" ) gap> 2u42:= CharacterTable( "2.U4(2)" );; gap> OrdersClassRepresentatives( 2u42 )[4]; 4 gap> GetFusionMap( 2u42, u42 )[4]; 3 gap> OrdersClassRepresentatives( u42 )[3]; 2 gap> List( PossibleClassFusions( u42, u ), x -> x[3] ); [ 8 ] gap> PositionsProperty( SizesConjugacyClasses( u ), x -> x = 3 ); [ 2 ] gap> ForAll( PossibleClassFusions( u, t ), x -> x[2] = x[8] ); true
The last subgroup for which the structure of the preimage has to be shown is \(U = (A_6 \times A_6):2^2\). We claim that each of the \(A_6\) type subgroups in the derived subgroup \(U^{\prime} = A_6 \times A_6\) lifts to its double cover in \(2.S\). Since all elements of order \(20\) in \(U\) lie in \(U^{\prime}\), at least one of the two direct factors must lift to its double cover, in order to give rise to an order \(40\) element in \(U\). In fact both factors lift to the double cover since the two direct factors are interchanged by conjugation in \(U\); the latter follows form tha fact that \(U\) has no normal subgroup of type \(A_6\).
gap> u:= CharacterTable( Maxes( t )[24] ); CharacterTable( "(A6xA6):2^2" ) gap> ClassPositionsOfDerivedSubgroup( u ); [ 1 .. 22 ] gap> PositionsProperty( OrdersClassRepresentatives( u ), x -> x = 20 ); [ 8 ] gap> List( ClassPositionsOfNormalSubgroups( u ), > x -> Sum( SizesConjugacyClasses( u ){ x } ) ); [ 1, 129600, 259200, 259200, 259200, 518400 ]
So statement (e) holds.
For statement (f), we have to consider the upward extensions \(S.2_1\), \(S.2_2\), and \(S.3\).
First we look at \(S.2_1\), an extension by an outer automorphism that acts as a double transposition in the outer automorphism group \(S_4\). Note that the symmetry between the three classes of element oder \(20\) in \(S\) is broken in \(S.2_1\), two of these classes have square roots in \(S.2_1\), the third has not.
gap> t2:= CharacterTable( "O8+(3).2_1" );; gap> ord20:= PositionsProperty( OrdersClassRepresentatives( t2 ), > x -> x = 20 );; gap> ord20:= Intersection( ord20, ClassPositionsOfDerivedSubgroup( t2 ) ); [ 84, 85, 86 ] gap> List( ord20, x -> x in PowerMap( t2, 2 ) ); [ false, true, true ]
Changing the viewpoint, we see that for each class of element order \(20\) in \(S\), there is a group of the type \(S.2_1\) in which the elements in this class do not have square roots, and there are groups of this type in which these elements have square roots. So we have to deal with two different cases, and we do this by first collecting the permutation characters induced from all maximal subgroups of \(S.2_1\) (other than \(S\)) that contain elements of order \(20\) in \(S\), and then considering \(s\) in each of these classes of \(S\).
We fix an embedding of \(S\) into \(S.2_1\) in which the elements in the class 20A do not have square roots. This situation is given for the stored class fusion between the tables in the GAP Character Table Library.
gap> tfust2:= GetFusionMap( t, t2 );; gap> tfust2{ PositionsProperty( OrdersClassRepresentatives( t ), > x -> x = 20 ) }; [ 84, 85, 86 ]
The six different actions of \(S\) on the cosets of \(O_7(3)\) type subgroups induce pairwise different permutation characters that form an orbit under the action of \(Aut(S)\). Four of these characters cannot extend to \(S.2_1\), the other two extend to permutation characters of \(S.2_1\) on the cosets of \(O_7(3).2\) type subgroups; these subgroups contain 20A elements.
gap> primt2:= [];; gap> poss:= PossiblePermutationCharacters( CharacterTable( "O7(3)" ), t );; gap> invfus:= InverseMap( tfust2 );; gap> List( poss, pi -> ForAll( CompositionMaps( pi, invfus ), IsInt ) ); [ false, false, false, false, true, true ] gap> PossiblePermutationCharacters( > CharacterTable( "O7(3)" ) * CharacterTable( "Cyclic", 2 ), t2 ); [ ] gap> ext:= PossiblePermutationCharacters( CharacterTable( "O7(3).2" ), t2 );; gap> List( ext, pi -> pi{ ord20 } ); [ [ 1, 0, 0 ], [ 1, 0, 0 ] ] gap> Append( primt2, ext );
The novelties in \(S.2_1\) that arise from \(O_7(3)\) type subgroups of \(S\) have the structure \(L_4(3).2^2\). These subgroups contain elements in the classes 20B and 20C of \(S\).
gap> ext:= PossiblePermutationCharacters( CharacterTable( "L4(3).2^2" ), t2 );; gap> List( ext, pi -> pi{ ord20 } ); [ [ 0, 0, 1 ], [ 0, 1, 0 ] ] gap> Append( primt2, ext );
Note that from the possible permutation characters of \(S.2_1\) on the cosets of \(L_4(3):2 \times 2\) type subgroups, we see that such subgroups must contain 20A elements, i. e., all such subgroups of \(S.2_1\) lie inside \(O_7(3).2\) type subgroups. This means that the structure description of these novelties in [CCN+85, p. 140] is not correct. The correct structure is \(L_4(3).2^2\).)
gap> List( PossiblePermutationCharacters( CharacterTable( "L4(3).2_2" ) * > CharacterTable( "Cyclic", 2 ), t2 ), pi -> pi{ ord20 } ); [ [ 1, 0, 0 ] ]
All \(3^6:L_4(3)\) type subgroups of \(S\) extend to \(S.2_1\). We compute these permutation characters as the possible permutation characters of the right degree.
gap> ext:= PermChars( t2, rec( torso:= [ 1120 ] ) );; gap> List( ext, pi -> pi{ ord20 } ); [ [ 2, 0, 0 ], [ 0, 0, 2 ], [ 0, 2, 0 ] ] gap> Append( primt2, ext );
Also all \(2.U_4(3).2^2\) type subgroups of \(S\) extend to \(S.2_1\). We compute the permutation characters as the extensions of the corresponding permutation characters of \(S\).
gap> filt:= Filtered( prim, x -> x[1] = 189540 );; gap> cand:= List( filt, x -> CompositionMaps( x, invfus ) );; gap> ext:= Concatenation( List( cand, > pi -> PermChars( t2, rec( torso:= pi ) ) ) );; gap> List( ext, x -> x{ ord20 } ); [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] gap> Append( primt2, ext );
The extensions of \((A_4 \times U_4(2)):2\) type subgroups of \(S\) to \(S.2_1\) have the type \(S_4 \times U_4(2):2\), they contain 20A elements.
gap> ext:= PossiblePermutationCharacters( CharacterTable( "Symmetric", 4 ) * > CharacterTable( "U4(2).2" ), t2 );; gap> List( ext, x -> x{ ord20 } ); [ [ 1, 0, 0 ], [ 1, 0, 0 ] ] gap> Append( primt2, ext );
All \((A_6 \times A_6):2^2\) type subgroups of \(S\) extend to \(S.2_1\). We compute the permutation characters as the extensions of the corresponding permutation characters of \(S\).
gap> filt:= Filtered( prim, x -> x[1] = 9552816 );; gap> cand:= List( filt, x -> CompositionMaps( x, InverseMap( tfust2 ) ));; gap> ext:= Concatenation( List( cand, > pi -> PermChars( t2, rec( torso:= pi ) ) ) );; gap> List( ext, x -> x{ ord20 } ); [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] gap> Append( primt2, ext );
We have found all relevant permutation characters of \(S.2_1\). This together with the list in [CCN+85, p. 140] implies statement (g).
Now we compute the bounds \(σ^{\prime}(S.2_1, s)\).
gap> Length( primt2 ); 15 gap> approx:= List( ord20, x -> ApproxP( primt2, x ) );; gap> outer:= Difference( > PositionsProperty( OrdersClassRepresentatives( t2 ), IsPrimeInt ), > ClassPositionsOfDerivedSubgroup( t2 ) );; gap> List( approx, l -> Maximum( l{ outer } ) ); [ 574/1215, 83/567, 83/567 ]
Next we look at \(S.2_2\), an extension by an outer automorphism that acts as a transposition in the outer automorphism group \(S_4\). Similar to the above situation, the symmetry between the three classes of element oder \(20\) in \(S\) is broken also in \(S.2_2\): The first is a conjugacy class of \(S.2_2\), the other two classes are fused in \(S.2_2\),
gap> t2:= CharacterTable( "O8+(3).2_2" );; gap> ord20:= PositionsProperty( OrdersClassRepresentatives( t2 ), > x -> x = 20 );; gap> ord20:= Intersection( ord20, ClassPositionsOfDerivedSubgroup( t2 ) ); [ 82, 83 ] gap> tfust2:= GetFusionMap( t, t2 );; gap> tfust2{ PositionsProperty( OrdersClassRepresentatives( t ), > x -> x = 20 ) }; [ 82, 83, 83 ]
Like in the case \(S.2_1\), we compute the permutation characters induced from all maximal subgroups of \(S.2_2\) (other than \(S\)) that contain elements of order \(20\) in \(S\).
We fix the embedding of \(S\) into \(S.2_2\) in which the class 20A of \(S\) is a class of \(S.2_2\). This situation is given for the stored class fusion between the tables in the GAP Character Table Library.
Exactly two classes of \(O_7(3)\) type subgroups in \(S\) extend to \(S.2_2\), these groups contain 20A elements.
gap> primt2:= [];; gap> ext:= PermChars( t2, rec( torso:= [ 1080 ] ) );; gap> List( ext, pi -> pi{ ord20 } ); [ [ 1, 0 ], [ 1, 0 ] ] gap> Append( primt2, ext );
Only one class of \(3^6:L_4(3)\) type subgroups extends to \(S.2_2\). (Note that we need not consider the novelties of the type \(3^{3+6}:(L_3(3) \times 2)\), because the order of these groups is not divisible by \(5\).)
gap> ext:= PermChars( t2, rec( torso:= [ 1120 ] ) );; gap> List( ext, pi -> pi{ ord20 } ); [ [ 2, 0 ] ] gap> Append( primt2, ext );
Only one class of \(2.U_4(3).2^2\) type subgroups of \(S\) extends to \(S.2_2\). We compute the permutation character as the extension of the corresponding permutation characters of \(S\).
gap> filt:= Filtered( prim, x -> x[1] = 189540 );; gap> cand:= List( filt, x -> CompositionMaps( x, InverseMap( tfust2 ) ));; gap> ext:= Concatenation( List( cand, > pi -> PermChars( t2, rec( torso:= pi ) ) ) );; gap> List( ext, x -> x{ ord20 } ); [ [ 1, 0 ] ] gap> Append( primt2, ext );
Two classes of \((A_4 \times U_4(2)):2\) type subgroups of \(S\) extend to \(S.2_2\).
gap> filt:= Filtered( prim, x -> x[1] = 7960680 );; gap> cand:= List( filt, x -> CompositionMaps( x, InverseMap( tfust2 ) ));; gap> ext:= Concatenation( List( cand, > pi -> PermChars( t2, rec( torso:= pi ) ) ) );; gap> List( ext, x -> x{ ord20 } ); [ [ 1, 0 ], [ 1, 0 ] ] gap> Append( primt2, ext );
Exactly one class of \((A_6 \times A_6):2^2\) type subgroups in \(S\) extends to \(S.2_2\), and the extensions have the structure \(S_6 \wr 2\).
gap> ext:= PossiblePermutationCharacters( CharacterTableWreathSymmetric( > CharacterTable( "S6" ), 2 ), t2 );; gap> List( ext, x -> x{ ord20 } ); [ [ 1, 0 ] ] gap> Append( primt2, ext );
We have found all relevant permutation characters of \(S.2_2\), and compute the bounds \(σ^{\prime}(S.2_2, s)\).
gap> Length( primt2 ); 7 gap> approx:= List( ord20, x -> ApproxP( primt2, x ) );; gap> outer:= Difference( > PositionsProperty( OrdersClassRepresentatives( t2 ), IsPrimeInt ), > ClassPositionsOfDerivedSubgroup( t2 ) );; gap> List( approx, l -> Maximum( l{ outer } ) ); [ 14/9, 0 ]
This means that there is an extension of the type \(S.2_2\) in which \(s\) cannot be chosen such that the bound is less than \(1/2\). More precisely, we have \(σ(g,s) \geq 1/2\) exactly for \(g\) in the unique outer involution class of size \(1\,080\).
gap> approx:= ApproxP( primt2, ord20[1] );; gap> bad:= Filtered( outer, i -> approx[i] >= 1/2 ); [ 84 ] gap> OrdersClassRepresentatives( t2 ){ bad }; [ 2 ] gap> SizesConjugacyClasses( t2 ){ bad }; [ 1080 ] gap> Number( SizesConjugacyClasses( t2 ), x -> x = 1080 ); 1
So we compute the proportion of elements in this class that generate \(S.2_2\) together with an element \(s\) of order \(20\) in \(S\). (As above, we have to consider two conjugacy classes.) For that, we first compute a permutation representation of \(S.2_2\), using that \(S.2_2\) is isomporphic to the two subgroups of index \(2\) in \(PGO^+(8,3) = O_8^+(3).2^2_{122}\) that are different from \(PSO^+(8,3) = O_8^+(3).2_1\), cf. [CCN+85, p. 140].
gap> go:= GO(1,8,3);; gap> so:= SO(1,8,3);; gap> outerelm:= First( GeneratorsOfGroup( go ), x -> not x in so );; gap> g2:= ClosureGroup( DerivedSubgroup( so ), outerelm );; gap> Size( g2 ); 19808719257600 gap> dom:= NormedRowVectors( GF(3)^8 );; gap> orbs:= OrbitsDomain( g2, dom, OnLines );; gap> List( orbs, Length ); [ 1080, 1080, 1120 ] gap> act:= Action( g2, orbs[1], OnLines );;
An involution \(g\) can be found as a power of one of the given generators.
gap> Order( outerelm ); 26 gap> g:= Permutation( outerelm^13, orbs[1], OnLines );; gap> Size( ConjugacyClass( act, g ) ); 1080
Now we find the candidates for the elements \(s\), and compute their ratios of nongeneration.
gap> ord20; [ 82, 83 ] gap> SizesCentralizers( t2 ){ ord20 }; [ 40, 20 ] gap> der:= DerivedSubgroup( act );; gap> repeat 20A:= Random( der ); > until Order( 20A ) = 20 and Size( Centralizer( act, 20A ) ) = 40; gap> RatioOfNongenerationTransPermGroup( act, g, 20A ); 1 gap> repeat 20BC:= Random( der ); > until Order( 20BC ) = 20 and Size( Centralizer( act, 20BC ) ) = 20; gap> RatioOfNongenerationTransPermGroup( act, g, 20BC ); 0
This means that for \(s\) in one \(S\)-class of elements of order \(20\), we have \(P^{\prime}(g, s) = 1\), and \(s\) in the other two \(S\)-classes of elements of order \(20\) generates with any conjugate of \(g\).
Concerning \(S.2_2\), it remains to show that we cannot find a better element than \(s\). For that, we first compute class representatives \(s^{\prime}\) in \(S\), w.r.t. conjugacy in \(S.2_2\), and then compute \(P^{\prime}( s^{\prime}, g )\). (It would be enough to check representatives of classes of maximal element order, but computing all classes is easy enough.)
gap> ccl:= ConjugacyClasses( act );; gap> der:= DerivedSubgroup( act );; gap> reps:= Filtered( List( ccl, Representative ), x -> x in der );; gap> Length( reps ); 83 gap> ratios:= List( reps, > s -> RatioOfNongenerationTransPermGroup( act, g, s ) );; gap> cand:= PositionsProperty( ratios, x -> x < 1 );; gap> ratios:= ratios{ cand };; gap> SortParallel( ratios, cand ); gap> ratios; [ 0, 1/10, 1/10, 16/135, 1/3, 1/3, 11/27, 7/15, 7/15 ]
For \(S.2_2\), it remains to show that there is no element \(s^{\prime} \in S\) such that \(P^{\prime}( {s^{\prime}}^x, g ) < 1\) holds for any \(x \in Aut(S)\) and \(g \in S.2_2\). So we are done when we can show that each class given by cand is conjugate in \(S.3\) to a class outside cand. The classes can be identified by element orders and centralizer orders.
gap> invs:= List( cand, > x -> [ Order( reps[x] ), Size( Centralizer( der, reps[x] ) ) ] ); [ [ 20, 20 ], [ 18, 108 ], [ 18, 108 ], [ 14, 28 ], [ 15, 45 ], [ 15, 45 ], [ 10, 40 ], [ 12, 72 ], [ 12, 72 ] ]
Namely, cand contains no full \(S.3\)-orbit of classes of the element orders \(20\), \(18\), \(14\), \(15\), and \(10\); also, cand does not contain full \(S.3\)-orbits on the classes 12O–12T.
Finally, we deal with \(S.3\). The fact that no maximal subgroup of \(S\) containing an element of order \(20\) extends to \(S.3\) follows either from the list of maximal subgroups of \(S\) in [CCN+85, p. 140] or directly from the permutation characters.
gap> t3:= CharacterTable( "O8+(3).3" );; gap> tfust3:= GetFusionMap( t, t3 );; gap> inv:= InverseMap( tfust3 );; gap> filt:= PositionsProperty( prim, x -> x[ spos ] <> 0 );; gap> ForAll( prim{ filt }, > pi -> ForAny( CompositionMaps( pi, inv ), IsList ) ); true
So we have to consider only the classes of novelties in \(S.3\), but the order of none of these groups is divisible by \(20\) –again see [CCN+85, p. 140]). This means that any element in \(S.3 \setminus S\) together with an element of order \(20\) in \(S\) generates \(S.3\). This is in fact stronger than statement (f), which claims this property only for elements of prime order in \(S.3 \setminus S\) (and their roots); note that \(S.3 \setminus S\) contains elements of the orders \(9\) and \(27\).
gap> outer:= Difference( [ 1 .. NrConjugacyClasses( t3 ) ], > ClassPositionsOfDerivedSubgroup( t3 ) ); [ 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94 ] gap> Set( OrdersClassRepresentatives( t3 ){ outer } ); [ 3, 6, 9, 12, 18, 21, 24, 27, 36, 39 ]
Before we turn to the next computations, we clean the workspace.
gap> CleanWorkspace();
We show that \(S = O^+_8(4) = \Omega^+(8,4)\) satisfies the following.
For suitable \(s \in S\) of the type \(2^- \perp 6^-\) (i. e., \(s\) decomposes the natural \(8\)-dimensional module for \(S\) into an orthogonal sum of two irreducible modules of the dimensions \(2\) and \(6\), respectively) and of order \(65\), \(𝕄(S,s)\) consists of exactly three pairwise nonconjugate subgroups of the type \((5 \times O^-_6(4)).2 = (5 \times \Omega^-(6,4)).2\).
\(σ( S, s ) \leq 34\,817 / 1\,645\,056\).
In the extensions \(S.2_1\) and \(S.3\) of \(S\) by graph automorphisms, there is at most one maximal subgroup besides \(S\) that contains \(s\). For the extension \(S.2_2\) of \(S\) by a field automorphism, we have \(σ^{\prime}(S.2_2, s) = 0\). In the extension \(S.2_3\) of \(S\) by the product of an involutory graph automorphism and a field automorphism, there is a unique maximal subgroup besides \(S\) that contains \(s\).
A safe source for determining \(𝕄(S,s)\) is [Kle87]. By inspection of the result matrix in this paper, we get that the only maximal subgroups of \(S\) that contain elements of order \(65\) occur in the rows 9–14 and 23–25; they have the isomorphism types \(S_6(4) = Sp(6,4) \cong O_7(4) = \Omega(7,4)\) and \((5 \times O_6^-(4)).2 = (5 \times \Omega^-(6,4)).2\), respectively, and for each of these, there are three conjugacy classes of subgroups in \(S\), which are conjugate under the triality graph automorphism of \(S\).
We start with the natural matrix representation of \(S\). For convenience, we compute an isomorphic permutation group on \(5\,525\) points.
gap> q:= 4;; n:= 8;; gap> G:= DerivedSubgroup( SO( 1, n, q ) );; gap> points:= NormedRowVectors( GF(q)^n );; gap> orbs:= OrbitsDomain( G, points, OnLines );; gap> List( orbs, Length ); [ 5525, 16320 ] gap> hom:= ActionHomomorphism( G, orbs[1], OnLines );; gap> G:= Image( hom );;
The group \(S\) contains exactly six conjugacy classes of (cyclic) subgroups of order \(65\); this follows from the fact that the centralizer of any Sylow \(13\) subgroup in \(S\) has the structure \(5 \times 5 \times 13\).
gap> Collected( Factors( Size( G ) ) ); [ [ 2, 24 ], [ 3, 5 ], [ 5, 4 ], [ 7, 1 ], [ 13, 1 ], [ 17, 2 ] ] gap> ResetGlobalRandomNumberGenerators(); gap> repeat x:= Random( G ); > until Order( x ) mod 13 = 0; gap> x:= x^( Order( x ) / 13 );; gap> c:= Centralizer( G, x );; gap> IsAbelian( c ); AbelianInvariants( c ); true [ 5, 5, 13 ]
The group \(S_6(4)\) contains exactly one class of subgroups of order \(65\), since the conjugacy classes of elements of order \(65\) in \(S_6(4)\) are algebraically conjugate.
gap> t:= CharacterTable( "S6(4)" );; gap> ord65:= PositionsProperty( OrdersClassRepresentatives( t ), > x -> x = 65 ); [ 105, 106, 107, 108, 109, 110, 111, 112 ] gap> ord65 = ClassOrbit( t, ord65[1] ); true
Thus there are at least three classes of order \(65\) elements in \(S\) that are not contained in \(S_6(4)\) type subgroups of \(S\). So we choose such an element \(s\), and have to consider only overgroups of the type \((5 \times \Omega^-(6,4)).2\).
The group \(\Omega^-(6,4) \cong U_4(4)\) contains exactly one class of subgroups of order \(65\).
gap> t:= CharacterTable( "U4(4)" );; gap> ords:= OrdersClassRepresentatives( t );; gap> ord65:= PositionsProperty( ords, x -> x = 65 );; gap> ord65 = ClassOrbit( t, ord65[1] ); true
So \(5 \times \Omega^-(6,4)\) contains exactly six such classes. Furthermore, subgroups in different classes are not \(S\)-conjugate.
gap> syl5:= SylowSubgroup( c, 5 );; gap> elms:= Filtered( Elements( syl5 ), y -> Order( y ) = 5 );; gap> reps:= Set( elms, SmallestGeneratorPerm );; Length( reps ); 6 gap> reps65:= List( reps, y -> SubgroupNC( G, [ y * x ] ) );; gap> pairs:= Combinations( reps65, 2 );; gap> ForAny( pairs, p -> IsConjugate( G, p[1], p[2] ) ); false
We consider only subgroups \(M \leq S\) in the three \(S\)-classes of the type \((5 \times \Omega^-(6,4)).2\).
gap> cand:= List( reps, y -> Normalizer( G, SubgroupNC( G, [ y ] ) ) );; gap> cand:= Filtered( cand, y -> Size( y ) = 10 * Size( t ) );; gap> Length( cand ); 3
(Note that one of the members in \(𝕄(S,s)\) is the stabilizer in \(S\) of the orthogonal decomposition \(2^- \perp 6^-\), the other two members are not reducible.)
By the above, the classes of subgroups of order \(65\) in each such \(M\) are in bijection with the corresponding classes in \(S\). Since \(N_S(\langle g \rangle) \subseteq M\) holds for any \(g \in M\) of order \(65\), also the conjugacy classes of elements of order \(65\) in \(M\) are in bijection with those in \(S\).
gap> norms:= List( reps65, y -> Normalizer( G, y ) );; gap> ForAll( norms, y -> ForAll( cand, M -> IsSubset( M, y ) ) ); true
As a consequence, we have \(g^S \cap M = g^M\) and thus \(1_M^S(g) = 1\). This implies statement (a).
In order to show statement (b), we want to use the function UpperBoundFixedPointRatios introduced in Section 11.3-3. For that, we first compute the conjugacy classes of the three class representatives \(M\). (Since the groups have elementary abelian Sylow \(5\) subgroups of the order \(5^4\), computing all conjugacy classes appears to be faster than using ClassesOfPrimeOrder.) Then we compute an upper bound for \(σ(S,s)\).
gap> syl5:= SylowSubgroup( cand[1], 5 );; gap> Size( syl5 ); IsElementaryAbelian( syl5 ); 625 true gap> UpperBoundFixedPointRatios( G, List( cand, ConjugacyClasses ), false ); [ 34817/1645056, false ]
Remark:
Computing the exact value \(σ(S,s)\) in the above setup would require to test the \(S\)-conjugacy of certain order \(5\) elements in \(M\). With the current GAP implementation, some of the relevant tests need several hours of CPU time.
An alternative approach would be to compute the permutation action of \(S\) on the cosets of \(M\), of degree \(6\,580\,224\), and to count the fixed points of conjugacy class representatives of prime order. The currently available GAP library methods are not sufficient for computing this in reasonable time. "Ad-hoc code" for this special case works, but it seemed to be not appropriate to include it here.
In the proof of statement (c), again we consult the result matrix in [Kle87]. For \(S.3\), the maximal subgroups are in the rows \(4\), \(15\), \(22\), \(26\), and \(61\). Only row \(26\) yields subgroups that contain elements \(s\) of order \(65\), they have the isomorphism type \((5 \times GU(3,4)).2 \cong (5^2 \times U_3(4)).2\). Note that the conjugacy classes of the members in \(𝕄(S,s)\) are permuted by the outer automorphism of order \(3\), so none of the subgroups in \(𝕄(S,s)\) extends to \(S.3\). By [BGK08, Lemma 2.4 (2)], if there is a maximal subgroup of \(S.3\) besides \(S\) that contains \(s\) then this subgroup is the normalizer in \(S.3\) of the intersection of the three members of \(𝕄(S,s)\), i. e., \(s\) is contained in at most one such subgroup.
For \(S.2_1\), only the rows \(9\) and \(23\) yield maximal subgroups containing elements of order \(65\), and since we had chosen \(s\) in such a way that row \(9\) was excluded already for the simple group, only extensions of the elements in \(𝕄(S,s)\) can appear. Exactly one of these three subgroups of \(S\) extends to \(S.2_1\), so again we get just one maximal subgroup of \(S.2_1\), besides \(S\), that contains \(s\).
All subgroups in \(𝕄(S,s)\) extend to \(S.2_2\), see [Kle87]. We compute the extensions of the above subgroups \(M\) of \(S\) to \(S.2_2\), by constructing the action of the field automorphism in the permutation representation we used for \(S\). In other words, we compute the projective action of the Frobenius map.
gap> frob:= PermList( List( orbs[1], v -> Position( orbs[1], > List( v, x -> x^2 ) ) ) );; gap> G2:= ClosureGroupDefault( G, frob );; gap> cand2:= List( cand, M -> Normalizer( G2, M ) );; gap> ccl:= List( cand2, > M2 -> PcConjugacyClassReps( SylowSubgroup( M2, 2 ) ) );; gap> List( ccl, l -> Number( l, x -> Order( x ) = 2 and not x in G ) ); [ 0, 0, 0 ]
So in each case, the extension of \(M\) to its normalizer in \(S.2_2\) is non-split. This implies \(σ^{\prime}(S.2_2,s) = 0\).
Finally, in the extension of \(S\) by the product of a graph automorphism and the field automorphism, exactly that member of \(𝕄(S,s)\) is invariant that is invariant under the graph automorphism, hence statement (c) holds.
It is again time to clean the workspace.
gap> CleanWorkspace();
The group \(S = O_9(3) = \Omega_9(3)\) is the first member in the series dealt with in [BGK08, Proposition 5.7], and serves as an example to illustrate this statement.
For \(s \in S\) of the type \(1 \perp 8^-\) (i. e., \(s\) decomposes the natural \(9\)-dimensional module for \(S\) into an orthogonal sum of two irreducible modules of the dimensions \(1\) and \(8\), respectively) and of order \((3^4 + 1)/2 = 41\), \(𝕄(S,s)\) consists of one group of the type \(O_8^-(3).2_1 = PGO^-(8,3)\).
\(σ(S,s) = 1/3\).
The uniform spread of \(S\) is at least three, with \(s\) of order \(41\).
By [MSW94], the only maximal subgroup of \(S\) that contains \(s\) is the stabilizer \(M\) of the orthogonal decomposition. The group \(2 \times O_8^-(3).2_1 = GO^-(8,3)\) embeds naturally into \(SO(9,3)\), its intersection with \(S\) is \(PGO^-(8,3)\). This proves statement (a).
The group \(M\) is the stabilizer of a \(1\)-space, it has index \(3\,240\) in \(S\).
gap> g:= SO( 9, 3 );; gap> g:= DerivedSubgroup( g );; gap> Size( g ); 65784756654489600 gap> orbs:= OrbitsDomain( g, NormedRowVectors( GF(3)^9 ), OnLines );; gap> List( orbs, Length ) / 41; [ 3240/41, 81, 80 ] gap> Size( SO( 9, 3 ) ) / Size( GO( -1, 8, 3 ) ); 3240
So we compute the unique transitive permutation character of \(S\) that has degree \(3\,240\).
gap> t:= CharacterTable( "O9(3)" );; gap> pi:= PermChars( t, rec( torso:= [ 3240 ] ) ); [ Character( CharacterTable( "O9(3)" ), [ 3240, 1080, 380, 132, 48, 324, 378, 351, 0, 0, 54, 27, 54, 27, 0, 118, 0, 36, 46, 18, 12, 2, 8, 45, 0, 108, 108, 135, 126, 0, 0, 56, 0, 0, 36, 47, 38, 27, 39, 36, 24, 12, 18, 18, 15, 24, 2, 18, 15, 9, 0, 0, 0, 2, 0, 18, 11, 3, 9, 6, 6, 9, 6, 3, 6, 3, 0, 6, 16, 0, 4, 6, 2, 45, 36, 0, 0, 0, 0, 0, 0, 0, 9, 9, 6, 3, 0, 0, 15, 13, 0, 5, 7, 36, 0, 10, 0, 10, 19, 6, 15, 0, 0, 0, 0, 12, 3, 10, 0, 3, 3, 7, 0, 6, 6, 2, 8, 0, 4, 0, 2, 0, 1, 3, 0, 0, 3, 0, 3, 2, 2, 3, 3, 6, 2, 2, 9, 6, 3, 0, 0, 18, 9, 0, 0, 12, 0, 0, 8, 0, 6, 9, 5, 0, 0, 0, 0, 0, 0, 0, 0, 3, 3, 3, 2, 1, 3, 3, 1, 0, 0, 4, 1, 0, 0, 1, 0, 3, 3, 1, 1, 2, 2, 0, 0, 1, 3, 4, 0, 1, 2, 0, 0, 1, 0, 4, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0 ] ) ] gap> spos:= Position( OrdersClassRepresentatives( t ), 41 ); 208 gap> approx:= ApproxP( pi, spos );; gap> Maximum( approx ); 1/3 gap> PositionsProperty( approx, x -> x = 1/3 ); [ 2 ] gap> SizesConjugacyClasses( t )[2]; 3321 gap> OrdersClassRepresentatives( t )[2]; 2
We see that \(P( S, s ) = σ( S, s ) = 1/3\) holds, and that \(σ( g, s )\) attains this maximum only for \(g\) in one class of involutions in \(S\); let us call this class 2A. (This class consists of the negatives of a class of reflections in \(GO(9,3)\).) This shows statement (b).
In order to show that the uniform spread of \(S\) is at least three, it suffices to show that for each triple of 2A elements, there is an element \(s\) of order \(41\) in \(S\) that generates \(S\) with each element of the triple.
We work with the primitive permutation representation of \(S\) on \(3\,240\) points. In this representation, \(s\) fixes exactly one point, and by statement (a), \(s\) generates \(S\) with \(x \in S\) if and only if \(x\) moves this point. Since the number of fixed points of each 2A involution in \(S\) is exactly one third of the moved points of \(S\), it suffices to show that we cannot choose three such involutions with mutually disjoint fixed point sets. And this is shown particularly easily because it will turn out that already for any two different 2A involutions, the sets of fixed points of are never disjoint.
First we compute a 2A element, which is determined as an involution with exactly \(1\,080\) fixed points.
gap> g:= Action( g, orbs[1], OnLines );; gap> repeat > repeat x:= Random( g ); ord:= Order( x ); until ord mod 2 = 0; > y:= x^(ord/2); > until NrMovedPoints( y ) = 3240 - 1080;
Next we compute the sets of fixed points of the elements in the class 2A, by forming the \(S\)-orbit of the set of fixed points of the chosen 2A element.
gap> fp:= Difference( MovedPoints( g ), MovedPoints( y ) );; gap> orb:= Orbit( g, fp, OnSets );;
Finally, we show that for any pair of 2A elements, their sets of fixed points intersect nontrivially. (Of course we can fix one of the two elements.) This proves statement (c).
gap> ForAny( orb, l -> IsEmpty( Intersection( l, fp ) ) ); false
We show that the group \(S = O_{10}^-(3) = PΩ^-(10,3)\) satisfies the following.
For \(s \in S\) irreducible of order \((3^5 + 1)/4 = 61\), \(𝕄(S,s)\) consists of one subgroup of the type \(SU(5,3) \cong U_5(3)\).
\(σ(S,s) = 1/1\,066\).
By [Ber00], the maximal subgroups of \(S\) containing \(s\) are of extension field type, and by [KL90, Prop. 4.3.18 and 4.3.20], these groups have the structure \(SU(5,3) = U_5(3)\) (which lift to \(2 \times U_5(3) < GU(5,3)\) in \(\Omega^-(10,3) = 2.S\)) or \(\Omega(5,9).2\), but the order of the latter group is not divisible by \(|s|\). Furthermore, by [BGK08, Lemma 2.12 (b)], \(s\) is contained in only one member of the former class.
gap> Size( GO(5,9) ) / 61; 6886425600/61
When the first version of these computations was written, the character tables of both \(S\) and \(U_5(3)\) were not contained in the GAP Character Table Library, so we worked with the groups. Meanwhile the character tables are available, thus we can show also a character theoretic solution.)
gap> t:= CharacterTable( "O10-(3)" ); s:= CharacterTable( "U5(3)" ); CharacterTable( "O10-(3)" ) CharacterTable( "U5(3)" ) gap> SigmaFromMaxes( t, "61A", [ s ], [ 1 ] ); 1/1066
(Now follow the computations with groups.)
The first step is the construction of the embedding of \(M = SU(5,3)\) into the matrix group \(2.S\), that is, we write the matrix generators of \(M\) as linear mappings on the natural module for \(2.S\), and then conjugate them such that the result matrices respect the bilinear form of \(2.S\). For convenience, we choose a basis for the field extension \(𝔽_9/𝔽_3\) such that the \(𝔽_3\)-linear mapping given by the invariant form of \(M\) is invariant under the \(𝔽_3\)-linear mappings given by the generators of \(M\).
gap> m:= SU(5,3);; gap> so:= SO(-1,10,3);; gap> omega:= DerivedSubgroup( so );; gap> om:= InvariantBilinearForm( so ).matrix;; gap> Display( om ); . 1 . . . . . . . . 1 . . . . . . . . . . . 1 . . . . . . . . . . 2 . . . . . . . . . . 2 . . . . . . . . . . 2 . . . . . . . . . . 2 . . . . . . . . . . 2 . . . . . . . . . . 2 . . . . . . . . . . 2 gap> b:= Basis( GF(9), [ Z(3)^0, Z(3^2)^2 ] ); Basis( GF(3^2), [ Z(3)^0, Z(3^2)^2 ] ) gap> blow:= List( GeneratorsOfGroup( m ), x -> BlownUpMat( b, x ) );; gap> form:= BlownUpMat( b, InvariantSesquilinearForm( m ).matrix );; gap> ForAll( blow, x -> x * form * TransposedMat( x ) = form ); true gap> Display( form ); . . . . . . . . 1 . . . . . . . . . . 1 . . . . . . 1 . . . . . . . . . . 1 . . . . . . 1 . . . . . . . . . . 1 . . . . . . 1 . . . . . . . . . . 1 . . . . . . 1 . . . . . . . . . . 1 . . . . . . . .
The matrix om of the invariant bilinear form of \(2.S\) is equivalent to the identity matrix \(I\). So we compute matrices T1 and T2 that transform om and form, respectively, to \(\pm I\).
gap> T1:= IdentityMat( 10, GF(3) );; gap> T1{[1..3]}{[1..3]}:= [[1,1,0],[1,-1,1],[1,-1,-1]]*Z(3)^0;; gap> pi:= PermutationMat( (1,10)(3,8), 10, GF(3) );; gap> tr:= NullMat( 10,10,GF(3) );; gap> tr{[1, 2]}{[1, 2]}:= [[1,1],[1,-1]]*Z(3)^0;; gap> tr{[3, 4]}{[3, 4]}:= [[1,1],[1,-1]]*Z(3)^0;; gap> tr{[7, 8]}{[7, 8]}:= [[1,1],[1,-1]]*Z(3)^0;; gap> tr{[9,10]}{[9,10]}:= [[1,1],[1,-1]]*Z(3)^0;; gap> tr{[5, 6]}{[5, 6]}:= [[1,0],[0,1]]*Z(3)^0;; gap> tr2:= IdentityMat( 10,GF(3) );; gap> tr2{[1,3]}{[1,3]}:= [[-1,1],[1,1]]*Z(3)^0;; gap> tr2{[7,9]}{[7,9]}:= [[-1,1],[1,1]]*Z(3)^0;; gap> T2:= tr2 * tr * pi;; gap> D:= T1^-1 * T2;; gap> tblow:= List( blow, x -> D * x * D^-1 );; gap> IsSubset( omega, tblow ); true
Now we switch to a permutation representation of \(S\), and use the embedding of \(M\) into \(2.S\) to obtain the corresponding subgroup of type \(M\) in \(S\). Then we compute an upper bound for \(\max\{ μ(g,S/M); g \in S^{\times} \}\).
gap> orbs:= OrbitsDomain( omega, NormedRowVectors( GF(3)^10 ), OnLines );; gap> List( orbs, Length ); [ 9882, 9882, 9760 ] gap> permgrp:= Action( omega, orbs[3], OnLines );; gap> M:= SubgroupNC( permgrp, > List( tblow, x -> Permutation( x, orbs[3], OnLines ) ) );; gap> ccl:= ClassesOfPrimeOrder( M, PrimeDivisors( Size( M ) ), > TrivialSubgroup( M ) );; gap> UpperBoundFixedPointRatios( permgrp, [ ccl ], false ); [ 1/1066, true ]
The entry true in the second position of the result indicates that in fact the exact value for the maximum of \(μ(g,S/M)\) has been computed. This implies statement (b).
We clean the workspace.
gap> CleanWorkspace();
We show that the group \(S = O_{14}^-(2) = \Omega^-(14,2)\) satisfies the following.
For \(s \in S\) irreducible of order \(2^7+1 = 129\), \(𝕄(S,s)\) consists of one subgroup \(M\) of the type \(GU(7,2) \cong 3 \times U_7(2)\).
\(σ(S,s) = 1/2\,015\).
By [Ber00], any maximal subgroup of \(S\) containing \(s\) is of extension field type, and by [KL90, Table 3.5.F, Prop. 4.3.18], these groups have the type \(GU(7,2)\), and there is exactly one class of subgroups of this type. Furthermore, by [BGK08, Lemma 2.12 (a)], \(s\) is contained in only one member of this class.
We embed \(U_7(2)\) into \(S\), by first replacing each element in \(𝔽_4\) by the \(2 \times 2\) matrix of the induced \(𝔽_2\)-linear mapping w.r.t. a suitable basis, and then conjugating the images of the generators such that the invariant quadratic form of \(S\) is respected.
gap> o:= SO(-1,14,2);; gap> g:= SU(7,2);; gap> b:= Basis( GF(4) );; gap> blow:= List( GeneratorsOfGroup( g ), x -> BlownUpMat( b, x ) );; gap> form:= NullMat( 14, 14, GF(2) );; gap> for i in [ 1 .. 14 ] do form[i][ 15-i ]:= Z(2); od; gap> ForAll( blow, x -> x * form * TransposedMat( x ) = form ); true gap> pi:= PermutationMat( (1,13)(3,11)(5,9), 14, GF(2) );; gap> pi * form * TransposedMat( pi ) = InvariantBilinearForm( o ).matrix; true gap> pi2:= PermutationMat( (7,3)(8,4), 14, GF(2) );; gap> D:= pi2 * pi;; gap> tblow:= List( blow, x -> D * x * D^-1 );; gap> IsSubset( o, tblow ); true
Note that the central subgroup of order three in \(GU(7,2)\) consists of scalar matrices.
gap> omega:= DerivedSubgroup( o );; gap> IsSubset( omega, tblow ); true gap> z:= Z(4) * One( g );; gap> tz:= D * BlownUpMat( b, z ) * D^-1;; gap> tz in omega; true
Now we switch to a permutation representation of \(S\), and compute the conjugacy classes of prime element order in the subgroup \(M\). The latter is done in two steps, first class representatives of the simple subgroup \(U_7(2)\) of \(M\) are computed, and then they are multiplied with the scalars in \(M\).
gap> orbs:= OrbitsDomain( omega, NormedRowVectors( GF(2)^14 ), OnLines );; gap> List( orbs, Length ); [ 8127, 8256 ] gap> omega:= Action( omega, orbs[1], OnLines );; gap> gens:= List( GeneratorsOfGroup( g ), > x -> Permutation( D * BlownUpMat( b, x ) * D^-1, orbs[1] ) );; gap> g:= Group( gens );; gap> ccl:= ClassesOfPrimeOrder( g, PrimeDivisors( Size( g ) ), > TrivialSubgroup( g ) );; gap> tz:= Permutation( tz, orbs[1] );; gap> primereps:= List( ccl, Representative );; gap> Add( primereps, () ); gap> reps:= Concatenation( List( primereps, > x -> List( [ 0 .. 2 ], i -> x * tz^i ) ) );; gap> primereps:= Filtered( reps, x -> IsPrimeInt( Order( x ) ) );; gap> Length( primereps ); 48
Finally, we apply UpperBoundFixedPointRatios (see Section 11.3-3) to compute an upper bound for \(μ(g,S/M)\), for \(g \in S^{\times}\).
gap> M:= ClosureGroup( g, tz );; gap> bccl:= List( primereps, x -> ConjugacyClass( M, x ) );; gap> UpperBoundFixedPointRatios( omega, [ bccl ], false ); [ 1/2015, true ]
Although some of the classes of \(M\) in the list bccl may be \(S\)-conjugate, the entry true in the second position of the result indicates that in fact the exact value for the maximum of \(μ(g,S/M)\), for \(g \in S^{\times}\), has been computed. This implies statement (b).
We clean the workspace.
gap> CleanWorkspace();
We show that the group \(S = O_{12}^+(3) = PΩ^+(12,3)\) satisfies the following.
\(S\) has a maximal subgroup \(M\) of the type \(N_S(PΩ^+(6,9))\), which has the structure \(PΩ^+(6,9).[4]\).
\(μ(g,S/M) \leq 2/88\,209\) holds for all \(g \in S^{\times}\).
(This result is used in the proof of [BGK08, Proposition 5.14], where it is shown that for \(s \in S\) of order \(205\), \(𝕄(S,s)\) consists of one reducible subgroup \(G_8\) and at most two extension field type subgroups of the type \(N_S(PΩ^+(6,9))\). By [GK00, Proposition 3.16], \(μ(g,S/G_8) \leq 19/3^5\) holds for all \(g \in S^{\times}\). This implies \(P(g,s) \leq 19/3^5 + 2 \cdot 2/88\,209 = 6\,901/88\,209 < 1/3\).)
Statement (a) follows from [KL90, Prop. 4.3.14].
For statement (b), we embed \(GO^+(6,9) \cong \Omega^+(6,9).2^2\) into \(SO^+(12,3) = 2.S.2\), by replacing each element in \(𝔽_9\) by the \(2 \times 2\) matrix of the induced \(𝔽_3\)-linear mapping w.r.t. a suitable basis \((b_1, b_2)\). We choose a basis with the property \(b_1 = 1\) and \(b_2^2 = 1 + b_2\), because then the image of a symmetric matrix is again symmetric (so the image of the invariant form is an invariant form for the image of the group), and apply an appropriate transformation to the images of the generators.
gap> so:= SO(+1,12,3);; gap> Display( InvariantBilinearForm( so ).matrix ); . 1 . . . . . . . . . . 1 . . . . . . . . . . . . . 1 . . . . . . . . . . . . 2 . . . . . . . . . . . . 2 . . . . . . . . . . . . 2 . . . . . . . . . . . . 2 . . . . . . . . . . . . 2 . . . . . . . . . . . . 2 . . . . . . . . . . . . 2 . . . . . . . . . . . . 2 . . . . . . . . . . . . 2 gap> g:= GO(+1,6,9);; gap> Z(9)^2 = Z(3)^0 + Z(9); true gap> b:= Basis( GF(9), [ Z(3)^0, Z(9) ] ); Basis( GF(3^2), [ Z(3)^0, Z(3^2) ] ) gap> blow:= List( GeneratorsOfGroup( g ), x -> BlownUpMat( b, x ) );; gap> m:= BlownUpMat( b, InvariantBilinearForm( g ).matrix );; gap> Display( m ); . . 1 . . . . . . . . . . . . 1 . . . . . . . . 1 . . . . . . . . . . . . 1 . . . . . . . . . . . . . . 2 . . . . . . . . . . . . 2 . . . . . . . . . . . . 2 . . . . . . . . . . . . 2 . . . . . . . . . . . . 2 . . . . . . . . . . . . 2 . . . . . . . . . . . . 2 . . . . . . . . . . . . 2 gap> pi:= PermutationMat( (2,3), 12, GF(3) );; gap> tr:= IdentityMat( 12, GF(3) );; gap> tr{[3,4]}{[3,4]}:= [[1,-1],[1,1]]*Z(3)^0;; gap> D:= tr * pi;; gap> D * m * TransposedMat( D ) = InvariantBilinearForm( so ).matrix; true gap> tblow:= List( blow, x -> D * x * D^-1 );; gap> IsSubset( so, tblow ); true
The image of \(GO^+(6,9)\) under the embedding into \(SO^+(12,3)\) does not lie in \(\Omega^+(12,3) = 2.S\), so a factor of two is missing in \(GO^+(6,9) \cap 2.S\) for getting (the preimage \(2.M\) of) the required maximal subgroup \(M\) of \(S\). Because of this, and also because currently it is time consuming to compute the derived subgroup of \(SO^+(12,3)\), we work with the upward extension \(PSO^+(12,3) = S.2\). Note that \(M\) extends to a maximal subgroup of \(S.2\).
First we factor out the centre of \(SO^+(12,3)\), and switch to a permutation representation of \(S.2\).
gap> orbs:= OrbitsDomain( so, NormedRowVectors( GF(3)^12 ), OnLines );; gap> List( orbs, Length ); [ 88452, 88452, 88816 ] gap> act:= Action( so, orbs[1], OnLines );; gap> SetSize( act, Size( so ) / 2 );
Next we rewrite the matrix generators for \(GO^+(6,9)\) accordingly, and compute the normalizer in \(S.2\) of the subgroup they generate; this is the maximal subgroup \(M.2\) we need.
gap> u:= SubgroupNC( act, > List( tblow, x -> Permutation( x, orbs[1], OnLines ) ) );; gap> n:= Normalizer( act, u );; gap> Size( n ) / Size( u ); 2
Now we compute class representatives of prime order in \(M.2\), in a smaller faithful permutation representation, and then the desired upper bound for \(μ(g, S/M)\).
gap> norbs:= OrbitsDomain( n, MovedPoints( n ) );; gap> List( norbs, Length ); [ 58968, 29484 ] gap> hom:= ActionHomomorphism( n, norbs[2] );; gap> nact:= Image( hom );; gap> Size( nact ) = Size( n ); true gap> ccl:= ClassesOfPrimeOrder( nact, PrimeDivisors( Size( nact ) ), > TrivialSubgroup( nact ) );; gap> Length( ccl ); 26 gap> preim:= List( ccl, > x -> PreImagesRepresentative( hom, Representative( x ) ) );; gap> pccl:= List( preim, x -> ConjugacyClass( n, x ) );; gap> for i in [ 1 .. Length( pccl ) ] do > SetSize( pccl[i], Size( ccl[i] ) ); > od; gap> UpperBoundFixedPointRatios( act, [ pccl ], false ); [ 2/88209, true ]
Note that we have computed \(\max\{ μ(g,S.2/M.2), g \in S.2^{\times} \} \geq \max\{ μ(g,S.2/M.2), g \in S^{\times} \} = \max\{ μ(g,S/M), g \in S^{\times} \}\).
We show that the group \(S = S_4(8) = Sp(4,8)\) satisfies the following.
For \(s \in S\) irreducible of order \(65\), \(𝕄(S,s)\) consists of two nonconjugate subgroups of the type \(S_2(64).2 = Sp(2,64).2 \cong L_2(64).2 \cong O_4^-(8).2 = \Omega^-(4,8).2\).
\(σ(S,s) = 8/63\).
By [Ber00], the only maximal subgroups of \(S\) that contain \(s\) are \(O_4^-(8).2 = SO^-(4,8)\) or of extension field type. By [KL90, Prop. 4.3.10, 4.8.6], there is one class of each of these subgroups (which happen to be isomorphic).
These classes of subgroups induce different permutation characters. One argument to see this is that the involutions in the outer half of extension field type subgroup \(S_2(64).2 < S_4(8)\) have a two-dimensional fixed space, whereas the outer involutions in \(SO^-(4,8)\) have a three-dimensional fixed space.
The former statement can be seen by using a normal basis of the field extension \(𝔽_{64}/𝔽_8\), such that the action of the Frobenius automorphism (which yields a suitable outer involution) is just a double transposition on the basis vectors of the natural module for \(S\).
gap> sp:= SP(4,8);; gap> Display( InvariantBilinearForm( sp ).matrix ); . . . 1 . . 1 . . 1 . . 1 . . . gap> z:= Z(64);; gap> f:= AsField( GF(8), GF(64) );; gap> repeat > b:= Basis( f, [ z, z^8 ] ); > z:= z * Z(64); > until b <> fail; gap> sub:= SP(2,64);; gap> Display( InvariantBilinearForm( sub ).matrix ); . 1 1 . gap> ext:= Group( List( GeneratorsOfGroup( sub ), > x -> BlownUpMat( b, x ) ) );; gap> tr:= PermutationMat( (3,4), 4, GF(2) );; gap> conj:= ConjugateGroup( ext, tr );; gap> IsSubset( sp, conj ); true gap> inv:= [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]] * Z(2);; gap> inv in sp; true gap> inv in conj; false gap> Length( NullspaceMat( inv - inv^0 ) ); 2
The latter statement can be shown by looking at an outer involution in \(SO^-(4,8)\).
gap> so:= SO(-1,4,8);; gap> der:= DerivedSubgroup( so );; gap> x:= First( GeneratorsOfGroup( so ), x -> not x in der );; gap> x:= x^( Order(x)/2 );; gap> Length( NullspaceMat( x - x^0 ) ); 3
The character table of \(L_2(64).2\) is currently not available in the GAP Character Table Library, so we compute the possible permutation characters with a combinatorial approach, and show statement (a).
gap> CharacterTable( "L2(64).2" ); fail gap> t:= CharacterTable( "S4(8)" );; gap> degree:= Size( t ) / ( 2 * Size( SL(2,64) ) );; gap> pi:= PermChars( t, rec( torso:= [ degree ] ) ); [ Character( CharacterTable( "S4(8)" ), [ 2016, 0, 256, 32, 0, 36, 0, 8, 1, 0, 4, 0, 0, 0, 28, 28, 28, 0, 0, 0, 0, 0, 0, 36, 36, 36, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 4, 4, 4, 0, 0, 0, 4, 4, 4, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ), Character( CharacterTable( "S4(8)" ), [ 2016, 256, 0, 32, 36, 0, 0, 8, 1, 4, 0, 28, 28, 28, 0, 0, 0, 0, 0, 0, 36, 36, 36, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 4, 4, 4, 0, 0, 0, 4, 4, 4, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ) ] gap> spos:= Position( OrdersClassRepresentatives( t ), 65 );; gap> List( pi, x -> x[ spos ] ); [ 1, 1 ]
Now we compute \(σ(S,s)\), which yields statement (b).
gap> Maximum( ApproxP( pi, spos ) ); 8/63
We clean the workspace.
gap> CleanWorkspace();
We show that the group \(S = S_6(2) = Sp(6,2)\) satisfies the following.
\(σ(S) = 4/7\), and this value is attained exactly for \(σ(S,s)\) with \(s\) of order \(9\).
For \(s \in S\) of order \(9\), \(𝕄(S,s)\) consists of one subgroup of the type \(U_4(2).2 = \Omega^-(6,2).2\) and three conjugate subgroups of the type \(L_2(8).3 = Sp(2,8).3\).
For \(s \in S\) of order \(9\), and \(g \in S^{\times}\), we have \(P(g,s) < 1/3\), except if \(g\) is in one of the classes 2A (the transvection class) or 3A.
For \(s \in S\) of order \(15\), and \(g \in S^{\times}\), we have \(P(g,s) < 1/3\), except if \(g\) is in one of the classes 2A or 2B.
\(P(S) = 11/21\), and this value is attained exactly for \(P(S,s)\) with \(s\) of order \(15\).
For all \(s^{\prime} \in S\), we have \(P(g,s^{\prime}) > 1/3\) for \(g\) in at least two classes.
The uniform spread of \(S\) is at least two, with \(s\) of order \(9\).
(Note that in this example, the optimal choice of \(s\) w.r.t. \(σ(S,s)\) is not optimal w.r.t. \(P(S,s)\).)
Statement (a) follows from the inspection of the primitive permutation characters, cf. Section 11.4-3.
gap> t:= CharacterTable( "S6(2)" );; gap> ProbGenInfoSimple( t ); [ "S6(2)", 4/7, 1, [ "9A" ], [ 4 ] ]
Also statement (b) follows from the information provided by the character table of \(S\) (cf. [CCN+85, p. 46]).
gap> prim:= PrimitivePermutationCharacters( t );; gap> ord:= OrdersClassRepresentatives( t );; gap> spos:= Position( ord, 9 );; gap> filt:= PositionsProperty( prim, x -> x[ spos ] <> 0 ); [ 1, 8 ] gap> Maxes( t ){ filt }; [ "U4(2).2", "L2(8).3" ] gap> List( prim{ filt }, x -> x[ spos ] ); [ 1, 3 ]
Now we consider statement (c). For \(s\) of order \(9\) and \(g\) in one of the classes 2A, 3A, we observe that \(P(g,s) = σ(g,s)\) holds. This is because exactly one maximal subgroup of \(S\) contains both \(s\) and \(g\). For all other elements \(g\), we have even \(σ(g,s) < 1/3\).
gap> prim:= PrimitivePermutationCharacters( t );; gap> spos9:= Position( ord, 9 );; gap> approx9:= ApproxP( prim, spos9 );; gap> filt9:= PositionsProperty( approx9, x -> x >= 1/3 ); [ 2, 6 ] gap> AtlasClassNames( t ){ filt9 }; [ "2A", "3A" ] gap> approx9{ filt9 }; [ 4/7, 5/14 ] gap> List( Filtered( prim, x -> x[ spos9 ] <> 0 ), x -> x{ filt9 } ); [ [ 16, 10 ], [ 0, 0 ] ]
Similarly, statement (d) follows. For \(s\) of order \(15\) and \(g\) in one of the classes 2A, 2B, already the degree \(36\) permutation character yields \(P(g,s) \geq 1/3\). And for all other elements \(g\), again we have \(σ(g,s) < 1/3\).
gap> spos15:= Position( ord, 15 );; gap> approx15:= ApproxP( prim, spos15 );; gap> filt15:= PositionsProperty( approx15, x -> x >= 1/3 ); [ 2, 3 ] gap> PositionsProperty( ApproxP( prim{ [ 2 ] }, spos15 ), x -> x >= 1/3 ); [ 2, 3 ] gap> AtlasClassNames( t ){ filt15 }; [ "2A", "2B" ] gap> approx15{ filt15 }; [ 46/63, 8/21 ]
For the remaining statements, we use explicit computations with \(S\), in the transitive degree \(63\) permutation representation. We start with a function that computes a transvection in \(S_d(2)\); note that the invariant bilinear form used for symplectic groups in GAP is described by a matrix with nonzero entries exactly in the positions \((i,d+1-i)\), for \(1 \leq i \leq d\).
gap> transvection:= function( d ) > local mat; > mat:= IdentityMat( d, Z(2) ); > mat{ [ 1, d ] }{ [ 1, d ] }:= [ [ 0, 1 ], [ 1, 0 ] ] * Z(2); > return mat; > end;;
First we compute, for statement (d), the exact values \(P(g,s)\) for \(g\) in one of the classes 2A or 2B, and \(s\) of order \(15\). Note that the classes 2A, 2B are the unique classes of the lengths \(63\) and \(315\), respectively.
gap> PositionsProperty( SizesConjugacyClasses( t ), x -> x in [ 63, 315 ] ); [ 2, 3 ] gap> d:= 6;; gap> matgrp:= Sp(d,2);; gap> hom:= ActionHomomorphism( matgrp, NormedRowVectors( GF(2)^d ) );; gap> g:= Image( hom, matgrp );; gap> ResetGlobalRandomNumberGenerators(); gap> repeat s15:= Random( g ); > until Order( s15 ) = 15; gap> 2A:= Image( hom, transvection( d ) );; gap> Size( ConjugacyClass( g, 2A ) ); 63 gap> IsTransitive( g, MovedPoints( g ) ); true gap> RatioOfNongenerationTransPermGroup( g, 2A, s15 ); 11/21 gap> repeat 12C:= Random( g ); > until Order( 12C ) = 12 and Size( Centralizer( g, 12C ) ) = 12; gap> 2B:= 12C^6;; gap> Size( ConjugacyClass( g, 2B ) ); 315 gap> RatioOfNongenerationTransPermGroup( g, 2B, s15 ); 8/21
For statement (e), we compute \(P(g, s^{\prime})\), for a transvection \(g\) and class representatives \(s^{\prime}\) of \(S\). It turns out that the minimum is \(11/21\), and it is attained for exactly one \(s^{\prime}\); by the above, this element has order \(15\).
gap> ccl:= ConjugacyClasses( g );; gap> reps:= List( ccl, Representative );; gap> nongen2A:= List( reps, > x -> RatioOfNongenerationTransPermGroup( g, 2A, x ) );; gap> min:= Minimum( nongen2A ); 11/21 gap> Number( nongen2A, x -> x = min ); 1
For statement (f), we show that for any choice of \(s^{\prime}\), at least two of the values \(P(g,s^{\prime})\), with \(g\) in the classes 2A, 2B, or 3A, are larger than \(1/3\).
gap> nongen2B:= List( reps, > x -> RatioOfNongenerationTransPermGroup( g, 2B, x ) );; gap> 3A:= s15^5;; gap> nongen3A:= List( reps, > x -> RatioOfNongenerationTransPermGroup( g, 3A, x ) );; gap> bad:= List( [ 1 .. NrConjugacyClasses( t ) ], > i -> Number( [ nongen2A, nongen2B, nongen3A ], > x -> x[i] > 1/3 ) );; gap> Minimum( bad ); 2
Finally, for statement (g), we have to consider only the case that the two elements \(x\), \(y\) are transvections.
gap> PositionsProperty( approx9, x -> x + approx9[2] >= 1 ); [ 2 ]
We use the random approach described in Section 11.3-3.
gap> repeat s9:= Random( g ); > until Order( s9 ) = 9; gap> RandomCheckUniformSpread( g, [ 2A, 2A ], s9, 20 ); true
We show that the group \(S = S_8(2)\) satisfies the following.
For \(s \in S\) of order \(17\), \(𝕄(S,s)\) consists of one subgroup of each of the types \(O_8^-(2).2 = \Omega^-(8,2).2\), \(S_4(4).2 = Sp(4,4).2\), and \(L_2(17) = PSL(2,17)\).
For \(s \in S\) of order \(17\), and \(g \in S^{\times}\), we have \(P(g,s) < 1/3\), except if \(g\) is a transvection.
The uniform spread of \(S\) is at least two, with \(s\) of order \(17\).
Statement (a) follows from the list of maximal subgroups of \(S\) in [CCN+85, p. 123], and the fact that \(1_H^S(s) = 1\) holds for each \(H \in 𝕄(S,s)\). Note that \(17\) divides the indices of the maximal subgroups of the types \(O_8^+(2).2\) and \(2^7 : S_6(2)\) in \(S\), and obviously \(17\) does not divide the orders of the remaining maximal subgroups.
The permutation characters induced from the first two subgroups are uniquely determined by the ordinary character tables. The permutation character induced from the last subgroup is uniquely determined if one considers also the corresponding Brauer tables; the correct class fusion is stored in the GAP Character Table Library, see [Brea].
gap> t:= CharacterTable( "S8(2)" );; gap> pi1:= PossiblePermutationCharacters( CharacterTable( "O8-(2).2" ), t );; gap> pi2:= PossiblePermutationCharacters( CharacterTable( "S4(4).2" ), t );; gap> pi3:= [ TrivialCharacter( CharacterTable( "L2(17)" ) )^t ];; gap> prim:= Concatenation( pi1, pi2, pi3 );; gap> Length( prim ); 3 gap> spos:= Position( OrdersClassRepresentatives( t ), 17 );; gap> List( prim, x -> x[ spos ] ); [ 1, 1, 1 ]
For statement (b), we observe that \(σ(g,s) < 1/3\) if \(g\) is not a transvection, and that \(P(g,s) = σ(g,s)\) for transvections \(g\) because exactly one of the three permutation characters is nonzero on both \(s\) and the class of transvections.
gap> approx:= ApproxP( prim, spos );; gap> PositionsProperty( approx, x -> x >= 1/3 ); [ 2 ] gap> Number( prim, pi -> pi[2] <> 0 and pi[ spos ] <> 0 ); 1 gap> approx[2]; 8/15
In statement (c), we have to consider only the case that the two elements \(x\), \(y\) are transvections.
gap> PositionsProperty( approx, x -> x + approx[2] >= 1 ); [ 2 ]
We use the random approach described in Section 11.3-3.
gap> d:= 8;; gap> matgrp:= Sp(d,2);; gap> hom:= ActionHomomorphism( matgrp, NormedRowVectors( GF(2)^d ) );; gap> x:= Image( hom, transvection( d ) );; gap> g:= Image( hom, matgrp );; gap> C:= ConjugacyClass( g, x );; Size( C ); 255 gap> ResetGlobalRandomNumberGenerators(); gap> repeat s:= Random( g ); > until Order( s ) = 17; gap> RandomCheckUniformSpread( g, [ x, x ], s, 20 ); true
We show that the group \(S = S_{10}(2)\) satisfies the following.
For \(s \in S\) of order \(33\), \(𝕄(S,s)\) consists of one subgroup of each of the types \(\Omega^-(10,2).2\) and \(L_2(32).5 = Sp(2,32).5\).
For \(s \in S\) of order \(33\), and \(g \in S^{\times}\), we have \(P(g,s) < 1/3\), except if \(g\) is a transvection.
The uniform spread of \(S\) is at least two, with \(s\) of order \(33\).
By [Ber00], the only maximal subgroups of \(S\) that contain \(s\) have the types stated in (a), and by [KL90, Prop. 4.3.10 and 4.8.6], there is exactly one class of each of these subgroups.
We compute the values \(σ( g, s )\), for all \(g \in S^{\times}\).
gap> t:= CharacterTable( "S10(2)" );; gap> pi1:= PossiblePermutationCharacters( CharacterTable( "O10-(2).2" ), t );; gap> pi2:= PossiblePermutationCharacters( CharacterTable( "L2(32).5" ), t );; gap> prim:= Concatenation( pi1, pi2 );; Length( prim ); 2 gap> spos:= Position( OrdersClassRepresentatives( t ), 33 );; gap> approx:= ApproxP( prim, spos );;
For statement (b), we observe that \(σ(g,s) < 1/3\) if \(g\) is not a transvection, and that \(P(g,s) = σ(g,s)\) for transvections \(g\) because exactly one of the two permutation characters is nonzero on both \(s\) and the class of transvections.
gap> PositionsProperty( approx, x -> x >= 1/3 ); [ 2 ] gap> Number( prim, pi -> pi[2] <> 0 and pi[ spos ] <> 0 ); 1 gap> approx[2]; 16/31
In statement (c), we have to consider only the case that the two elements \(x\), \(y\) are transvections. We use the random approach described in Section 11.3-3.
gap> d:= 10;; gap> matgrp:= Sp(d,2);; gap> hom:= ActionHomomorphism( matgrp, NormedRowVectors( GF(2)^d ) );; gap> x:= Image( hom, transvection( d ) );; gap> g:= Image( hom, matgrp );; gap> C:= ConjugacyClass( g, x );; Size( C ); 1023 gap> ResetGlobalRandomNumberGenerators(); gap> repeat s:= Random( g ); > until Order( s ) = 33; gap> RandomCheckUniformSpread( g, [ x, x ], s, 20 ); true
We show that \(S = U_4(2) = SU(4,2) \cong S_4(3) = PSp(4,3)\) satisfies the following.
\(σ(S) = 21/40\), and this value is attained exactly for \(σ(S,s)\) with \(s\) of order \(12\).
For \(s \in S\) of order \(9\), \(𝕄(S,s)\) consists of two groups, of the types \(3^{1+2}_+ \colon 2A_4 = GU(3,2)\) and \(3^3 \colon S_4\), respectively.
\(P(S) = 2/5\), and this value is attained exactly for \(P(S,s)\) with \(s\) of order \(9\).
The uniform spread of \(S\) is at least three, with \(s\) of order \(9\).
\(σ^{\prime}(Aut(S),s) = 7/20\).
(Note that in this example, the optimal choice of \(s\) w.r.t. \(σ(S,s)\) is not optimal w.r.t. \(P(S,s)\).)
Statement (a) follows from inspection of the primitive permutation characters, cf. Section 11.4-3.
gap> t:= CharacterTable( "U4(2)" );; gap> ProbGenInfoSimple( t ); [ "U4(2)", 21/40, 1, [ "12A" ], [ 2 ] ]
Statement (b) can be read off from the permutation characters, and the fact that the only classes of maximal subgroups that contain elements of order \(9\) consist of groups of the structures \(3^{1+2}_+:2A_4\) and \(3^3:S_4\), see [CCN+85, p. 26].
gap> OrdersClassRepresentatives( t ); [ 1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 6, 6, 6, 6, 6, 6, 9, 9, 12, 12 ] gap> prim:= PrimitivePermutationCharacters( t ); [ Character( CharacterTable( "U4(2)" ), [ 27, 3, 7, 0, 0, 9, 0, 3, 1, 2, 0, 0, 3, 3, 0, 1, 0, 0, 0, 0 ] ), Character( CharacterTable( "U4(2)" ), [ 36, 12, 8, 0, 0, 6, 3, 0, 2, 1, 0, 0, 0, 0, 3, 2, 0, 0, 0, 0 ] ), Character( CharacterTable( "U4(2)" ), [ 40, 8, 0, 13, 13, 4, 4, 4, 0, 0, 5, 5, 2, 2, 2, 0, 1, 1, 1, 1 ] ), Character( CharacterTable( "U4(2)" ), [ 40, 16, 4, 4, 4, 1, 7, 0, 2, 0, 4, 4, 1, 1, 1, 1, 1, 1, 0, 0 ] ), Character( CharacterTable( "U4(2)" ), [ 45, 13, 5, 9, 9, 6, 3, 1, 1, 0, 1, 1, 4, 4, 1, 2, 0, 0, 1, 1 ] ) ]
For statement (c), we use a primitive permutation representation on \(40\) points that occurs in the natural action of \(SU(4,2)\).
gap> g:= SU(4,2);; gap> orbs:= OrbitsDomain( g, NormedRowVectors( GF(4)^4 ), OnLines );; gap> List( orbs, Length ); [ 45, 40 ] gap> g:= Action( g, orbs[2], OnLines );;
First we show that for \(s\) of order \(9\), \(P(S,s) = 2/5\) holds. For that, we have to consider only \(P(g,s)\), with \(g\) in one of the classes 2A (of length \(45\)) and 3A (of length \(40\)); since the class 3B contains the inverses of the elements in the class 3A, we need not test it.
gap> spos:= Position( OrdersClassRepresentatives( t ), 9 ); 17 gap> approx:= ApproxP( prim, spos ); [ 0, 3/5, 1/10, 17/40, 17/40, 1/8, 11/40, 1/10, 1/20, 0, 9/40, 9/40, 3/40, 3/40, 3/40, 1/40, 1/20, 1/20, 1/40, 1/40 ] gap> badpos:= PositionsProperty( approx, x -> x >= 2/5 ); [ 2, 4, 5 ] gap> PowerMap( t, 2 )[4]; 5 gap> OrdersClassRepresentatives( t ); [ 1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 6, 6, 6, 6, 6, 6, 9, 9, 12, 12 ] gap> SizesConjugacyClasses( t ); [ 1, 45, 270, 40, 40, 240, 480, 540, 3240, 5184, 360, 360, 720, 720, 1440, 2160, 2880, 2880, 2160, 2160 ]
A representative \(g\) of a class of length \(40\) can be found as the third power of any order \(9\) element.
gap> PowerMap( t, 3 )[ spos ]; 4 gap> ResetGlobalRandomNumberGenerators(); gap> repeat s:= Random( g ); > until Order( s ) = 9; gap> Size( ConjugacyClass( g, s^3 ) ); 40 gap> prop:= RatioOfNongenerationTransPermGroup( g, s^3, s ); 13/40
Next we examine \(g\) in the class 2A.
gap> repeat x:= Random( g ); until Order( x ) = 12; gap> Size( ConjugacyClass( g, x^6 ) ); 45 gap> prop:= RatioOfNongenerationTransPermGroup( g, x^6, s ); 2/5
Finally, we compute that for \(s\) of order different from \(9\) and \(g\) in the class 2A, \(P(g,s)\) is larger than \(2/5\).
gap> ccl:= List( ConjugacyClasses( g ), Representative );; gap> SortParallel( List( ccl, Order ), ccl ); gap> List( ccl, Order ); [ 1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 6, 6, 6, 6, 6, 6, 9, 9, 12, 12 ] gap> prop:= List( ccl, r -> RatioOfNongenerationTransPermGroup( g, x^6, r ) ); [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 5/9, 1, 1, 1, 1, 1, 1, 2/5, 2/5, 7/15, 7/15 ] gap> Minimum( prop ); 2/5
In order to show statement (d), we have to consider triples \((x_1, x_2, x_3)\) with \(x_i\) of prime order and \(\sum_{i=1}^3 P(x_i,s) \geq 1\). This means that it suffices to check \(x\) in the class 2A, \(y\) in 2A\( \cup \)3A, and \(z\) in 2A\( \cup \)3A\( \cup \)3D.
gap> approx[2]:= 2/5;; gap> approx[4]:= 13/40;; gap> primeord:= PositionsProperty( OrdersClassRepresentatives( t ), > IsPrimeInt ); [ 2, 3, 4, 5, 6, 7, 10 ] gap> RemoveSet( primeord, 5 ); gap> primeord; [ 2, 3, 4, 6, 7, 10 ] gap> approx{ primeord }; [ 2/5, 1/10, 13/40, 1/8, 11/40, 0 ] gap> AtlasClassNames( t ){ primeord }; [ "2A", "2B", "3A", "3C", "3D", "5A" ] gap> triples:= Filtered( UnorderedTuples( primeord, 3 ), > t -> Sum( approx{ t } ) >= 1 ); [ [ 2, 2, 2 ], [ 2, 2, 4 ], [ 2, 2, 7 ], [ 2, 4, 4 ], [ 2, 4, 7 ] ]
We use the random approach described in Section 11.3-3.
gap> repeat 6E:= Random( g ); > until Order( 6E ) = 6 and Size( Centralizer( g, 6E ) ) = 18; gap> 2A:= 6E^3;; gap> 3A:= s^3;; gap> 3D:= 6E^2;; gap> RandomCheckUniformSpread( g, [ 2A, 2A, 2A ], s, 50 ); true gap> RandomCheckUniformSpread( g, [ 2A, 2A, 3A ], s, 50 ); true gap> RandomCheckUniformSpread( g, [ 3D, 2A, 2A ], s, 50 ); true gap> RandomCheckUniformSpread( g, [ 2A, 3A, 3A ], s, 50 ); true gap> RandomCheckUniformSpread( g, [ 3D, 3A, 2A ], s, 50 ); true
Statement (e) can be proved using ProbGenInfoAlmostSimple, cf. Section 11.4-4.
gap> t:= CharacterTable( "U4(2)" );; gap> t2:= CharacterTable( "U4(2).2" );; gap> spos:= PositionsProperty( OrdersClassRepresentatives( t ), x -> x = 9 );; gap> ProbGenInfoAlmostSimple( t, t2, spos ); [ "U4(2).2", 7/20, [ "9AB" ], [ 2 ] ]
We show that \(S = U_4(3) = PSU(4,3)\) satisfies the following.
\(σ(S) = 53/153\), and this value is attained exactly for \(σ(S,s)\) with \(s\) of order \(7\).
For \(s \in S\) of order \(7\), \(𝕄(S,s)\) consists of two nonconjugate groups of the type \(L_3(4)\), one group of the type \(U_3(3)\), and four pairwise nonconjugate groups of the type \(A_7\).
\(P(S) = 43/135\), and this value is attained exactly for \(P(S,s)\) with \(s\) of order \(7\).
The uniform spread of \(S\) is at least three, with \(s\) of order \(7\).
The preimage of \(s\) in the matrix group \(SU(4,3) \cong 4.U_4(3)\) has order \(28\), the preimages of the groups in \(𝕄(S,s)\) have the structures \(4_2.L_3(4)\), \(4 \times U_3(3) \cong GU(3,3)\), and \(4.A_7\) (the latter being a central product of a cyclic group of order four and \(2.A_7\)).
\(P^{\prime}(S.2_1,s) = 13/27\), \(σ^{\prime}(S.2_2) = 1/3\), and \(σ^{\prime}(S.2_3) = 31/162\), with \(s\) of order \(7\) in each case.
Statement (a) follows from inspection of the primitive permutation characters, cf. Section 11.4-3.
gap> t:= CharacterTable( "U4(3)" );; gap> ProbGenInfoSimple( t ); [ "U4(3)", 53/135, 2, [ "7A" ], [ 7 ] ]
Statement (b) can be read off from the permutation characters, and the fact that the only classes of maximal subgroups that contain elements of order \(7\) consist of groups of the structures as claimed, see [CCN+85, p. 52].
gap> prim:= PrimitivePermutationCharacters( t );; gap> spos:= Position( OrdersClassRepresentatives( t ), 7 ); 13 gap> List( Filtered( prim, x -> x[ spos ] <> 0 ), l -> l{ [ 1, spos ] } ); [ [ 162, 1 ], [ 162, 1 ], [ 540, 1 ], [ 1296, 1 ], [ 1296, 1 ], [ 1296, 1 ], [ 1296, 1 ] ]
In order to show statement (c) (which then implies statement (d)), we use a permutation representation on \(112\) points. It corresponds to an orbit of one-dimensional subspaces in the natural module of \(\Omega^-(6,3) \cong S\).
gap> matgrp:= DerivedSubgroup( SO( -1, 6, 3 ) );; gap> orbs:= OrbitsDomain( matgrp, NormedRowVectors( GF(3)^6 ), OnLines );; gap> List( orbs, Length ); [ 126, 126, 112 ] gap> G:= Action( matgrp, orbs[3], OnLines );;
It is sufficient to compute \(P(g,s)\), for involutions \(g \in S\).
gap> approx:= ApproxP( prim, spos ); [ 0, 53/135, 1/10, 1/24, 1/24, 7/45, 4/45, 1/27, 1/36, 1/90, 1/216, 1/216, 7/405, 7/405, 1/270, 0, 0, 0, 0, 1/270 ] gap> Filtered( approx, x -> x >= 43/135 ); [ 53/135 ] gap> OrdersClassRepresentatives( t ); [ 1, 2, 3, 3, 3, 3, 4, 4, 5, 6, 6, 6, 7, 7, 8, 9, 9, 9, 9, 12 ] gap> ResetGlobalRandomNumberGenerators(); gap> repeat g:= Random( G ); until Order(g) = 2; gap> repeat s:= Random( G ); > until Order(s) = 7; gap> bad:= RatioOfNongenerationTransPermGroup( G, g, s ); 43/135 gap> bad < 1/3; true
Statement (e) can be shown easily with character-theoretic methods, as follows. Since \(SU(4,3)\) is a Schur cover of \(S\) and the groups in \(𝕄(S,s)\) are simple, only very few possibilities have to be checked. The Schur multiplier of \(U_3(3)\) is trivial (see, e. g., [CCN+85, p. 14]), so the preimage in \(SU(4,3)\) is a direct product of \(U_3(3)\) and the centre of \(SU(4,3)\). Neither \(L_3(4)\) nor its double cover \(2.L_3(4)\) can be a subgroup of \(SU(4,3)\), so the preimage of \(L_3(4)\) must be a Schur cover of \(L_3(4)\), i. e., it must have either the type \(4_1.L_3(4)\) or \(4_2.L_3(4)\) (see [CCN+85, p. 23]); only the type \(4_2.L_3(4)\) turns out to be possible.
gap> 4t:= CharacterTable( "4.U4(3)" );; gap> Length( PossibleClassFusions( CharacterTable( "L3(4)" ), 4t ) ); 0 gap> Length( PossibleClassFusions( CharacterTable( "2.L3(4)" ), 4t ) ); 0 gap> Length( PossibleClassFusions( CharacterTable( "4_1.L3(4)" ), 4t ) ); 0 gap> Length( PossibleClassFusions( CharacterTable( "4_2.L3(4)" ), 4t ) ); 4
As for the preimage of the \(A_7\) type subgroups, we first observe that the double cover of \(A_7\) cannot be a subgroup of the double cover of \(S\), so the preimage of \(A_7\) in the double cover of \(U_4(3)\) is a direct product \(2 \times A_7\). The group \(SU(4,3)\) does not contain \(A_7\) type subgroups, thus the \(A_7\) type subgroups in \(2.U_4(3)\) lift to double covers of \(A_7\) in \(SU(4,3)\). This proves the claimed structure.
gap> 2t:= CharacterTable( "2.U4(3)" );; gap> Length( PossibleClassFusions( CharacterTable( "2.A7" ), 2t ) ); 0 gap> Length( PossibleClassFusions( CharacterTable( "A7" ), 4t ) ); 0
For statement (f), we consider automorphic extensions of \(S\). The bound for \(S.2_3\) has been computed in Section 11.4-4. That for \(S.2_2\) can be computed form the fact that the classes of maximal subgroups of \(S.2_2\) containing \(s\) of order \(7\) are \(S\), one class of \(U_3(3).2\) type subgroups, and two classes of \(S_7\) type subgroups which induce the same permutation character (see [CCN+85, p. 52]).
gap> t2:= CharacterTable( "U4(3).2_2" );; gap> pi1:= PossiblePermutationCharacters( CharacterTable( "U3(3).2" ), t2 ); [ Character( CharacterTable( "U4(3).2_2" ), [ 540, 12, 54, 0, 0, 9, 8, 0, 0, 6, 0, 0, 1, 2, 0, 0, 0, 2, 0, 24, 4, 0, 0, 0, 0, 0, 0, 3, 2, 0, 4, 0, 0, 0 ] ) ] gap> pi2:= PossiblePermutationCharacters( CharacterTable( "A7.2" ), t2 ); [ Character( CharacterTable( "U4(3).2_2" ), [ 1296, 48, 0, 27, 0, 9, 0, 4, 1, 0, 3, 0, 1, 0, 0, 0, 0, 0, 216, 24, 0, 4, 0, 0, 0, 9, 0, 3, 0, 1, 0, 1, 0, 0 ] ) ] gap> prim:= Concatenation( pi1, pi2, pi2 );; gap> outer:= Difference( > PositionsProperty( OrdersClassRepresentatives( t2 ), IsPrimeInt ), > ClassPositionsOfDerivedSubgroup( t2 ) );; gap> spos:= Position( OrdersClassRepresentatives( t2 ), 7 );; gap> Maximum( ApproxP( prim, spos ){ outer } ); 1/3
Finally, Section 11.4-4 shows that the character tables are not sufficient for what we need, so we compute the exact proportion of nongeneration for \(U_4(3).2_1 \cong SO^-(6,3)\).
gap> matgrp:= SO( -1, 6, 3 ); SO(-1,6,3) gap> orbs:= OrbitsDomain( matgrp, NormedRowVectors( GF(3)^6 ), OnLines );; gap> List( orbs, Length ); [ 126, 126, 112 ] gap> G:= Action( matgrp, orbs[3], OnLines );; gap> repeat s:= Random( G ); > until Order( s ) = 7; gap> repeat > repeat 2B:= Random( G ); until Order( 2B ) mod 2 = 0; > 2B:= 2B^( Order( 2B ) / 2 ); > c:= Centralizer( G, 2B ); > until Size( c ) = 12096; gap> RatioOfNongenerationTransPermGroup( G, 2B, s ); 13/27 gap> repeat > repeat 2C:= Random( G ); until Order( 2C ) mod 2 = 0; > 2C:= 2C^( Order( 2C ) / 2 ); > c:= Centralizer( G, 2C ); > until Size( c ) = 1440; gap> RatioOfNongenerationTransPermGroup( G, 2C, s ); 0
We show that \(S = U_6(3) = PSU(6,3)\) satisfies the following.
For \(s \in S\) of the type \(1 \perp 5\) (i. e., the preimage of \(s\) in \(2.S = SU(6,3)\) decomposes the natural \(6\)-dimensional module for \(2.S\) into an orthogonal sum of two irreducible modules of the dimensions \(1\) and \(5\), respectively) and of order \((3^5 + 1)/2 = 122\), \(𝕄(S,s)\) consists of one group of the type \(2 \times U_5(3)\), which lifts to a subgroup of the type \(4 \times U_5(3) = GU(5,3)\) in \(2.S\). (The preimage of \(s\) in \(2.S\) has order \(3^5 + 1 = 244\).)
\(σ(S,s) = 353/3\,159\).
By [MSW94], the only maximal subgroup of \(S\) that contains \(s\) is the stabilizer \(H \cong 2 \times U_5(3)\) of the orthogonal decomposition. This proves statement (a).
The character table of \(S\) is currently not available in the GAP Character Table Library. We consider the permutation action of \(S\) on the orbit of the stabilized \(1\)-space. So \(M\) can be taken as a point stabilizer in this action.
gap> CharacterTable( "U6(3)" ); fail gap> g:= SU(6,3);; gap> orbs:= OrbitsDomain( g, NormedRowVectors( GF(9)^6 ), OnLines );; gap> List( orbs, Length ); [ 22204, 44226 ] gap> repeat x:= PseudoRandom( g ); until Order( x ) = 244; gap> List( orbs, o -> Number( o, v -> OnLines( v, x ) = v ) ); [ 0, 1 ] gap> g:= Action( g, orbs[2], OnLines );; gap> M:= Stabilizer( g, 1 );;
Then we compute a list of elements in \(M\) that covers the conjugacy classes of prime element order, from which the numbers of fixed points and thus \(\max\{ μ( S/M, g ); g \in M^{\times} \} = σ( S, s )\) can be derived. This way we avoid completely to check the \(S\)-conjugacy of elements (class representatives of Sylow subgroups in \(M\)).
gap> elms:= [];; gap> for p in PrimeDivisors( Size( M ) ) do > syl:= SylowSubgroup( M, p ); > Append( elms, Filtered( PcConjugacyClassReps( syl ), > r -> Order( r ) = p ) ); > od; gap> 1 - Minimum( List( elms, NrMovedPoints ) ) / Length( orbs[2] ); 353/3159
We show that \(S = U_8(2) = SU(8,2)\) satisfies the following.
For \(s \in S\) of the type \(1 \perp 7\) (i. e., \(s\) decomposes the natural \(8\)-dimensional module for \(S\) into an orthogonal sum of two irreducible modules of the dimensions \(1\) and \(7\), respectively) and of order \(2^7 + 1 = 129\), \(𝕄(S,s)\) consists of one group of the type \(3 \times U_7(2) = GU(7,2)\).
\(σ(S,s) = 2\,753/10\,880\).
By [MSW94], the only maximal subgroup of \(S\) that contains \(s\) is the stabilizer \(M \cong GU(7,2)\) of the orthogonal decomposition. This proves statement (a).
The character table of \(S\) is currently not available in the GAP Character Table Library. We proceed exactly as in Section 11.5-25 in order to prove statement (b).
gap> CharacterTable( "U8(2)" ); fail gap> g:= SU(8,2);; gap> orbs:= OrbitsDomain( g, NormedRowVectors( GF(4)^8 ), OnLines );; gap> List( orbs, Length ); [ 10965, 10880 ] gap> repeat x:= PseudoRandom( g ); until Order( x ) = 129; gap> List( orbs, o -> Number( o, v -> OnLines( v, x ) = v ) ); [ 0, 1 ] gap> g:= Action( g, orbs[2], OnLines );; gap> M:= Stabilizer( g, 1 );; gap> elms:= [];; gap> for p in PrimeDivisors( Size( M ) ) do > syl:= SylowSubgroup( M, p ); > Append( elms, Filtered( PcConjugacyClassReps( syl ), > r -> Order( r ) = p ) ); > od; gap> Length( elms ); 611 gap> 1 - Minimum( List( elms, NrMovedPoints ) ) / Length( orbs[2] ); 2753/10880
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