GAP (ref) - Chapter 39: Groups

This chapter explains how to create groups and defines operations for groups, that is operations whose definition does not depend on the representation used. However methods for these operations in most cases will make use of the representation.

If not otherwise specified, in all examples in this chapter the group g will be the symmetric group $S_4$ acting on the letters $\{ 1, \ldots, 4 \}$.

39.1 Group Elements

Groups in GAP are written multiplicatively. The elements from which a group can be generated must permit multiplication and multiplicative inversion (see 31.14).

gap> a:=(1,2,3);;b:=(2,3,4);;
gap> One(a);
()
gap> Inverse(b);
(2,4,3)
gap> a*b;
(1,3)(2,4)
gap> Order(a*b);
2
gap> Order( [ [ 1, 1 ], [ 0, 1 ] ] );
infinity

The next example may run into an infinite loop because the given matrix in fact has infinite order.

gap> Order( [ [ 1, 1 ], [ 0, 1 ] ] * Indeterminate( Rationals ) );
#I  Order: warning, order of <mat> might be infinite

Since groups are domains, the recommended command to compute the order of a group is Size (30.4-6). For convenience, group orders can also be computed with Order (31.10-10).

The operation Comm (31.12-3) can be used to compute the commutator of two elements, the operation LeftQuotient (31.12-2) computes the product $x^{{-1}} y$.

39.2 Creating Groups

When groups are created from generators, this means that the generators must be elements that can be multiplied and inverted (see also 31.3). For creating a free group on a set of symbols, see FreeGroup (37.2-1).

39.2-1 Group

If the only argument gens is a list that is not a matrix then Group( gens ) is the group generated by the elements of that list.

If there are two arguments, a list gens and an element id, then Group( gens, id ) is the group generated by the elements of gens, with identity id.

Note that the value of the attribute GeneratorsOfGroup (39.2-4) need not be equal to the list gens of generators entered as argument. Use GroupWithGenerators (39.2-3) if you want to be sure that the argument gens is stored as value of GeneratorsOfGroup (39.2-4).

gap> g:=Group((1,2,3,4),(1,2));
Group([ (1,2,3,4), (1,2) ])

39.2-2 GroupByGenerators

GroupByGenerators returns the group $G$ generated by the list gens. If a second argument id is present then this is stored as the identity element of the group.

The value of the attribute GeneratorsOfGroup (39.2-4) of $G$ need not be equal to gens. GroupByGenerators is the underlying operation called by Group (39.2-1).

39.2-3 GroupWithGenerators

GroupWithGenerators returns the group $G$ generated by the list gens. If a second argument id is present then this is stored as the identity element of the group. The value of the attribute GeneratorsOfGroup (39.2-4) of $G$ is equal to gens.

39.2-4 GeneratorsOfGroup

returns a list of generators of the group G. If G has been created by the command GroupWithGenerators (39.2-3) with argument gens, then the list returned by GeneratorsOfGroup will be equal to gens. For such a group, each generator can also be accessed using the . operator (see GeneratorsOfDomain (31.9-2)): for a positive integer $i$, G.i returns the $i$-th element of the list returned by GeneratorsOfGroup. Moreover, if G is a free group, and name is the name of a generator of G then G.name also returns this generator.

gap> g:=GroupWithGenerators([(1,2,3,4),(1,2)]);
Group([ (1,2,3,4), (1,2) ])
gap> GeneratorsOfGroup(g);
[ (1,2,3,4), (1,2) ]

While in this example GAP displays the group via the generating set stored in the attribute GeneratorsOfGroup, the methods installed for View (6.3-3) will in general display only some information about the group which may even be just the fact that it is a group.

39.2-5 AsGroup

if the elements of the collection D form a group the command returns this group, otherwise it returns fail.

gap> AsGroup([(1,2)]);
fail
gap> AsGroup([(),(1,2)]);
Group([ (1,2) ])

39.2-6 ConjugateGroup

returns the conjugate group of G, obtained by applying the conjugating element obj.

To form a conjugate (group) by any object acting via ^, one can also use the infix operator ^.

gap> ConjugateGroup(g,(1,5));
Group([ (2,3,4,5), (2,5) ])

39.2-7 IsGroup

A group is a magma-with-inverses (see IsMagmaWithInverses (35.1-4)) and associative (see IsAssociative (35.4-7)) multiplication.

IsGroup tests whether the object obj fulfills these conditions, it does not test whether obj is a set of elements that forms a group under multiplication; use AsGroup (39.2-5) if you want to perform such a test. (See 13.3 for details about categories.)

gap> IsGroup(g);
true

39.2-8 InfoGroup

39.3 Subgroups

For the general concept of parents and subdomains, see 31.7 and 31.8. More functions that construct certain subgroups can be found in the sections 39.11, 39.12, 39.13, and 39.14.

If a group $U$ is created as a subgroup of another group $G$, $G$ becomes the parent of $U$. There is no universal parent group, parent-child chains can be arbitrary long. GAP stores the result of some operations (such as Normalizer (39.11-1)) with the parent as an attribute.

39.3-1 Subgroup

creates the subgroup U of G generated by gens. The Parent (31.7-1) value of U will be G. The NC version does not check, whether the elements in gens actually lie in G.

The unary version of Subgroup creates a (shell) subgroup that does not even know generators but can be used to collect information about a particular subgroup over time.

gap> u:=Subgroup(g,[(1,2,3),(1,2)]);
Group([ (1,2,3), (1,2) ])

39.3-2 Index (GAP operation)

For a subgroup U of the group G, Index returns the index $[\textit{G}:\textit{U}] = |\textit{G}| / |\textit{U}|$ of U in G. The NC version does not test whether U is contained in G.

39.3-3 IndexInWholeGroup

If the family of elements of G itself forms a group P, this attribute returns the index of G in P. It is used primarily for free groups or finitely presented groups.

gap> freegp:=FreeGroup(1);;
gap> freesub:=Subgroup(freegp,[freegp.1^5]);;
gap> IndexInWholeGroup(freesub);
5

39.3-4 AsSubgroup

gap> v:=AsSubgroup(g,Group((1,2,3),(1,4)));
Group([ (1,2,3), (1,4) ])
gap> Parent(v);
Group([ (1,2,3,4), (1,2) ])

39.3-5 IsSubgroup

IsSubgroup returns true if U is a group that is a subset of the domain G. This is actually checked by calling IsGroup( U ) and IsSubset( G, U ); note that special methods for IsSubset (30.5-1) are available that test only generators of U if G is closed under the group operations. So in most cases, for example whenever one knows already that U is a group, it is better to call only IsSubset (30.5-1).

gap> IsSubgroup(g,u);
true
gap> v:=Group((1,2,3),(1,2));
Group([ (1,2,3), (1,2) ])
gap> u=v;
true
gap> IsSubgroup(g,v);
true

39.3-6 IsNormal

A group G normalizes a group U if and only if for every $g \in \textit{G}$ and $u \in \textit{U}$ the element $u^g$ is a member of U. Note that U need not be a subgroup of G.

gap> IsNormal(g,u);
false

39.3-7 IsCharacteristicSubgroup

gap> IsCharacteristicSubgroup(g,u);
false

39.3-8 ConjugateSubgroup

For a group G which has a parent group P (see Parent (31.7-1)), returns the subgroup of P, obtained by conjugating G using the conjugating element g.

If G has no parent group, it just delegates to the call to ConjugateGroup (39.2-6) with the same arguments.

To form a conjugate (subgroup) by any object acting via ^, one can also use the infix operator ^.

39.3-9 ConjugateSubgroups

39.3-10 IsSubnormal

A subgroup U of the group G is subnormal if it is contained in a subnormal series of G.

gap> IsSubnormal(g,Group((1,2,3)));
false
gap> IsSubnormal(g,Group((1,2)(3,4)));
true

39.3-11 SubgroupByProperty

creates a subgroup of G consisting of those elements fulfilling prop (which is a tester function). No test is done whether the property actually defines a subgroup.

Note that currently very little functionality beyond an element test exists for groups created this way.

39.3-12 SubgroupShell

creates a subgroup of G which at this point is not yet specified further (but will be later, for example by assigning a generating set).

gap> u:=SubgroupByProperty(g,i->3^i=3);
<subgrp of Group([ (1,2,3,4), (1,2) ]) by property>
gap> (1,3) in u; (1,4) in u; (1,5) in u;
false
true
false
gap> GeneratorsOfGroup(u);
[ (1,2), (1,4,2) ]
gap> u:=SubgroupShell(g);
<group>

39.4 Closures of (Sub)groups

39.4-1 ClosureGroup

creates the group generated by the elements of G and obj. obj can be either an element or a collection of elements, in particular another group.

gap> g:=SmallGroup(24,12);;u:=Subgroup(g,[g.3,g.4]);
Group([ f3, f4 ])
gap> ClosureGroup(u,g.2);
Group([ f2, f3, f4 ])
gap> ClosureGroup(u,[g.1,g.2]);
Group([ f1, f2, f3, f4 ])
gap> ClosureGroup(u,Group(g.2*g.1));
Group([ f1*f2^2, f3, f4 ])

39.4-2 ClosureGroupAddElm

These three functions together with ClosureGroupDefault (39.4-3) implement the main methods for ClosureGroup (39.4-1). In the ordering given, they just add elm to the generators, remove duplicates and identity elements, and test whether elm is already contained in G.

39.4-3 ClosureGroupDefault

This functions returns the closure of the group G with the element elm. If G has the attribute AsSSortedList (30.3-10) then also the result has this attribute. This is used to implement the default method for Enumerator (30.3-2) and EnumeratorSorted (30.3-3).

39.4-4 ClosureSubgroup

For a group G that stores a parent group (see 31.7), ClosureSubgroup calls ClosureGroup (39.4-1) with the same arguments; if the result is a subgroup of the parent of G then the parent of G is set as parent of the result, otherwise an error is raised. The check whether the result is contained in the parent of G is omitted by the NC version. As a wrong parent might imply wrong properties this version should be used with care.

39.5 Expressing Group Elements as Words in Generators

Using homomorphisms (see chapter 40) it is possible to express group elements as words in given generators: Create a free group (see FreeGroup (37.2-1)) on the correct number of generators and create a homomorphism from this free group onto the group G in whose generators you want to factorize. Then the preimage of an element of G is a word in the free generators, that will map on this element again.

39.5-1 EpimorphismFromFreeGroup

For a group G with a known generating set, this attribute returns a homomorphism from a free group that maps the free generators to the generators of G.

The option names can be used to prescribe a (print) name for the free generators.

The following example shows how to decompose elements of $S_4$ in the generators (1,2,3,4) and (1,2):

gap> g:=Group((1,2,3,4),(1,2));
Group([ (1,2,3,4), (1,2) ])
gap> hom:=EpimorphismFromFreeGroup(g:names:=["x","y"]);
[ x, y ] -> [ (1,2,3,4), (1,2) ]
gap> PreImagesRepresentative(hom,(1,4));
y^-1*x^-1*(x^-1*y^-1)^2*x

The following example stems from a real request to the GAP Forum. In September 2000 a GAP user working with puzzles wanted to express the permutation (1,2) as a word as short as possible in particular generators of the symmetric group $S_{16}$.

gap> perms := [ (1,2,3,7,11,10,9,5), (2,3,4,8,12,11,10,6),
>   (5,6,7,11,15,14,13,9), (6,7,8,12,16,15,14,10) ];;
gap> puzzle := Group( perms );;Size( puzzle );
20922789888000
gap> hom:=EpimorphismFromFreeGroup(puzzle:names:=["a", "b", "c", "d"]);;
gap> word := PreImagesRepresentative( hom, (1,2) );
a^-1*c*b*c^-1*a*b^-1*a^-2*c^-1*a*b^-1*c*b
gap> Length( word );
13

39.5-2 Factorization

returns a factorization of elm as word in the generators of the group G given in the attribute GeneratorsOfGroup (39.2-4). The attribute EpimorphismFromFreeGroup (39.5-1) of G will contain a map from the free group in which the word is expressed to the group G. The attribute MappingGeneratorsImages (40.10-2) of this map gives a list of generators and corresponding letters.

The algorithm used forms all elements of the group to ensure a short word is found. Therefore this function should not be used when the group G has more than a few million elements. Because of this, one should not call this function within algorithms, but use homomorphisms instead.

gap> G:=SymmetricGroup( 6 );;
gap> r:=(3,4);; s:=(1,2,3,4,5,6);;
gap> # create subgroup to force the system to use the generators r and s:
gap> H:= Subgroup(G, [ r, s ] );
Group([ (3,4), (1,2,3,4,5,6) ])
gap> Factorization( H, (1,2,3) );
(x2*x1)^2*x2^-2
gap> s*r*s*r*s^-2;
(1,2,3)
gap> MappingGeneratorsImages(EpimorphismFromFreeGroup(H));
[ [ x1, x2 ], [ (3,4), (1,2,3,4,5,6) ] ]

39.5-3 GrowthFunctionOfGroup

For a group G with a generating set given in GeneratorsOfGroup (39.2-4), this function calculates the number of elements whose shortest expression as words in the generating set is of a particular length. It returns a list L, whose $i+1$ entry counts the number of elements whose shortest word expression has length $i$. If a maximal length radius is given, only words up to length radius are counted. Otherwise the group must be finite and all elements are enumerated.

gap> GrowthFunctionOfGroup(MathieuGroup(12));
[ 1, 5, 19, 70, 255, 903, 3134, 9870, 25511, 38532, 16358, 382 ]
gap> GrowthFunctionOfGroup(MathieuGroup(12),2);
[ 1, 5, 19 ]
gap> GrowthFunctionOfGroup(MathieuGroup(12),99);
[ 1, 5, 19, 70, 255, 903, 3134, 9870, 25511, 38532, 16358, 382 ]
gap> free:=FreeGroup("a","b");
<free group on the generators [ a, b ]>
gap> product:=free/ParseRelators(free,"a2,b3");
<fp group on the generators [ a, b ]>
gap> SetIsFinite(product,false);
gap> GrowthFunctionOfGroup(product,10);
[ 1, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64 ]

39.6 Structure Descriptions

39.6-1 StructureDescription

The method for StructureDescription exhibits a structure of the given group G to some extent, using the strategy outlined below. The idea is to return a possibly short string which gives some insight in the structure of the considered group. It is intended primarily for small groups (order less than 100) or groups with few normal subgroups, in other cases, in particular large $p$-groups, it can be very costly. Furthermore, the string returned is -- as the action on chief factors is not described -- often not the most useful way to describe a group.

The string returned by StructureDescription is not an isomorphism invariant: non-isomorphic groups can have the same string value, and two isomorphic groups in different representations can produce different strings. The value returned by StructureDescription is a string of the following form:

  StructureDescription(<G>) ::=
      1                                 ; trivial group
    | C<size>                           ; finite cyclic group
    | Z                                 ; infinite cyclic group
    | A<degree>                         ; alternating group
    | S<degree>                         ; symmetric group
    | D<size>                           ; dihedral group
    | Q<size>                           ; quaternion group
    | QD<size>                          ; quasidihedral group
    | PSL(<n>,<q>)                      ; projective special linear group
    | SL(<n>,<q>)                       ; special linear group
    | GL(<n>,<q>)                       ; general linear group
    | PSU(<n>,<q>)                      ; proj. special unitary group
    | O(2<n>+1,<q>)                     ; orthogonal group, type B
    | O+(2<n>,<q>)                      ; orthogonal group, type D
    | O-(2<n>,<q>)                      ; orthogonal group, type 2D
    | PSp(2<n>,<q>)                     ; proj. special symplectic group
    | Sz(<q>)                           ; Suzuki group
    | Ree(<q>)                          ; Ree group (type 2F or 2G)
    | E(6,<q>) | E(7,<q>) | E(8,<q>)    ; Lie group of exceptional type
    | 2E(6,<q>) | F(4,<q>) | G(2,<q>)
    | 3D(4,<q>)                         ; Steinberg triality group
    | M11 | M12 | M22 | M23 | M24
    | J1 | J2 | J3 | J4 | Co1 | Co2
    | Co3 | Fi22 | Fi23 | Fi24' | Suz
    | HS | McL | He | HN | Th | B
    | M | ON | Ly | Ru                  ; sporadic simple group
    | 2F(4,2)'                          ; Tits group
    | PerfectGroup(<size>,<id>)         ; the indicated group from the
                                        ; library of perfect groups
    | A x B                             ; direct product
    | N : H                             ; semidirect product
    | C(G) . G/C(G) = G' . G/G'         ; non-split extension
                                        ; (equal alternatives and
                                        ; trivial extensions omitted)
    | Phi(G) . G/Phi(G)                 ; non-split extension:
                                        ; Frattini subgroup and
                                        ; Frattini factor group

Note that the StructureDescription is only one possible way of building up the given group from smaller pieces.

The option short is recognized - if this option is set, an abbreviated output format is used (e.g. "6x3" instead of "C6 x C3").

If the Name (12.8-2) attribute is not bound, but StructureDescription is, View (6.3-3) prints the value of the attribute StructureDescription. The Print (6.3-4)ed representation of a group is not affected by computing a StructureDescription.

Note that StructureDescription is not intended to be a research tool, but rather an educational tool. The reasons for this are as follows:

gap> l := AllSmallGroups(12);;
gap> List(l,StructureDescription);; l;
[ C3 : C4, C12, A4, D12, C6 x C2 ]
gap> List(AllSmallGroups(40),G->StructureDescription(G:short));
[ "5:8", "40", "5:8", "5:Q8", "4xD10", "D40", "2x(5:4)", "(10x2):2",
  "20x2", "5xD8", "5xQ8", "2x(5:4)", "2^2xD10", "10x2^2" ]
gap> List(AllTransitiveGroups(DegreeAction,6),
>         G->StructureDescription(G:short));
[ "6", "S3", "D12", "A4", "3xS3", "2xA4", "S4", "S4", "S3xS3",
  "(3^2):4", "2xS4", "A5", "(S3xS3):2", "S5", "A6", "S6" ]
gap> StructureDescription(SmallGroup(504,7));
"C7 : (C9 x Q8)"
gap> StructureDescription(SmallGroup(504,7):nice);
"(C7 : Q8) : C9"
gap> StructureDescription(AbelianGroup([0,2,3]));
"Z x C6"
gap> StructureDescription(AbelianGroup([0,0,0,2,3,6]):short);
"Z^3x6^2"
gap> StructureDescription(PSL(4,2));
"A8"

39.7 Cosets

39.7-1 RightCoset

returns the right coset of U with representative g, which is the set of all elements of the form $ug$ for all $u \in \textit{U}$. g must be an element of a larger group G which contains U. For element operations such as in a right coset behaves like a set of group elements.

Right cosets are external orbits for the action of U which acts via OnLeftInverse (41.2-3). Of course the action of a larger group G on right cosets is via OnRight (41.2-2).

gap> u:=Group((1,2,3), (1,2));;
gap> c:=RightCoset(u,(2,3,4));
RightCoset(Group( [ (1,2,3), (1,2) ] ),(2,3,4))
gap> ActingDomain(c);
Group([ (1,2,3), (1,2) ])
gap> Representative(c);
(2,3,4)
gap> Size(c);
6
gap> AsList(c);
[ (2,3,4), (1,4,2), (1,3,4,2), (1,3)(2,4), (2,4), (1,4,2,3) ]
gap> IsBiCoset(c);
false

39.7-2 RightCosets

computes a duplicate free list of right cosets U $g$ for $g \in$ G. A set of representatives for the elements in this list forms a right transversal of U in G. (By inverting the representatives one obtains a list of representatives of the left cosets of U.) The NC version does not check whether U is a subgroup of G.

gap> RightCosets(g,u);
[ RightCoset(Group( [ (1,2,3), (1,2) ] ),()),
  RightCoset(Group( [ (1,2,3), (1,2) ] ),(1,3)(2,4)),
  RightCoset(Group( [ (1,2,3), (1,2) ] ),(1,4)(2,3)),
  RightCoset(Group( [ (1,2,3), (1,2) ] ),(1,2)(3,4)) ]

39.7-3 CanonicalRightCosetElement

returns a canonical representative of the right coset U g which is independent of the given representative g. This can be used to compare cosets by comparing their canonical representatives.

The representative chosen to be the canonical one is representation dependent and only guaranteed to remain the same within one GAP session.

gap> CanonicalRightCosetElement(u,(2,4,3));
(3,4)

39.7-4 IsRightCoset

GAP does not provide left cosets as a separate data type, but as the left coset $gU$ consists of exactly the inverses of the elements of the right coset $Ug^{{-1}}$ calculations with left cosets can be emulated using right cosets by inverting the representatives.

39.7-5 IsBiCoset

A (right) coset $Ug$ is considered a bicoset if its set of elements simultaneously forms a left coset for the same subgroup. This is the case if and only if the coset representative $g$ normalizes the subgroup $U$.

39.7-6 CosetDecomposition

For a finite group G and a subgroup $\textit{S} \leq \textit{G}$ this function returns a partition of the elements of G according to the (right) cosets of S. The result is a list of lists, each sublist corresponding to one coset. The first sublist is the elements list of the subgroup, the other lists are arranged accordingly.

gap> CosetDecomposition(SymmetricGroup(4),SymmetricGroup(3));
[ [ (), (2,3), (1,2), (1,2,3), (1,3,2), (1,3) ],
  [ (1,4), (1,4)(2,3), (1,2,4), (1,2,3,4), (1,3,2,4), (1,3,4) ],
  [ (1,4,2), (1,4,2,3), (2,4), (2,3,4), (1,3)(2,4), (1,3,4,2) ],
  [ (1,4,3), (1,4,3,2), (1,2,4,3), (1,2)(3,4), (2,4,3), (3,4) ] ]

39.8 Transversals

39.8-1 RightTransversal

A right transversal $t$ is a list of representatives for the set $\textit{U} \setminus \textit{G}$ of right cosets (consisting of cosets $Ug$) of $U$ in $G$.

The object returned by RightTransversal is not a plain list, but an object that behaves like an immutable list of length $[\textit{G}:\textit{U}]$, except if U is the trivial subgroup of G in which case RightTransversal may return the sorted plain list of coset representatives.

The operation PositionCanonical (21.16-3), called for a transversal $t$ and an element $g$ of G, will return the position of the representative in $t$ that lies in the same coset of U as the element $g$ does. (In comparison, Position (21.16-1) will return fail if the element is not equal to the representative.) Functions that implement group actions such as Action (41.7-2) or Permutation (41.9-1) (see Chapter 41) use PositionCanonical (21.16-3), therefore it is possible to act on a right transversal to implement the action on the cosets. This is often much more efficient than acting on cosets.

gap> g:=Group((1,2,3,4),(1,2));;
gap> u:=Subgroup(g,[(1,2,3),(1,2)]);;
gap> rt:=RightTransversal(g,u);
RightTransversal(Group([ (1,2,3,4), (1,2) ]),Group([ (1,2,3), (1,2) ]))
gap> Length(rt);
4
gap> Position(rt,(1,2,3));
fail

Note that the elements of a right transversal are not necessarily canonical in the sense of CanonicalRightCosetElement (39.7-3), but we may compute a list of canonical coset representatives by calling that function. (See also PositionCanonical (21.16-3).)

gap> List(RightTransversal(g,u),i->CanonicalRightCosetElement(u,i));
[ (), (2,3,4), (1,2,3,4), (3,4) ]
gap> PositionCanonical(rt,(1,2,3));
1
gap> rt[1];
()

39.9 Double Cosets

39.9-1 DoubleCoset

The groups U and V must be subgroups of a common supergroup G of which g is an element. This command constructs the double coset U g V which is the set of all elements of the form $ugv$ for any $u \in \textit{U}$, $v \in \textit{V}$. For element operations such as in, a double coset behaves like a set of group elements. The double coset stores U in the attribute LeftActingGroup, g as Representative (30.4-7), and V as RightActingGroup.

39.9-2 RepresentativesContainedRightCosets

A double coset $\textit{D} = U g V$ can be considered as a union of right cosets $U h_i$. (It is the union of the orbit of $U g$ under right multiplication by $V$.) For a double coset D this function returns a set of representatives $h_i$ such that D $= \bigcup_{{h_i}} U h_i$. The representatives returned are canonical for $U$ (see CanonicalRightCosetElement (39.7-3)) and form a set.

gap> u:=Subgroup(g,[(1,2,3),(1,2)]);;v:=Subgroup(g,[(3,4)]);;
gap> c:=DoubleCoset(u,(2,4),v);
DoubleCoset(Group( [ (1,2,3), (1,2) ] ),(2,4),Group( [ (3,4) ] ))
gap> (1,2,3) in c;
false
gap> (2,3,4) in c;
true
gap> LeftActingGroup(c);
Group([ (1,2,3), (1,2) ])
gap> RightActingGroup(c);
Group([ (3,4) ])
gap> RepresentativesContainedRightCosets(c);
[ (2,3,4) ]

39.9-3 DoubleCosets

computes a duplicate free list of all double cosets U $g$ V for $g \in \textit{G}$. The groups U and V must be subgroups of the group G. The NC version does not check whether U and V are subgroups of G.

gap> dc:=DoubleCosets(g,u,v);
[ DoubleCoset(Group( [ (1,2,3), (1,2) ] ),(),Group( [ (3,4) ] )),
  DoubleCoset(Group( [ (1,2,3), (1,2) ] ),(1,3)(2,4),Group(
    [ (3,4) ] )), DoubleCoset(Group( [ (1,2,3), (1,2) ] ),(1,4)
    (2,3),Group( [ (3,4) ] )) ]
gap> List(dc,Representative);
[ (), (1,3)(2,4), (1,4)(2,3) ]

39.9-4 IsDoubleCoset

39.9-5 DoubleCosetRepsAndSizes

returns a list of double coset representatives and their sizes, the entries are lists of the form $[ r, n ]$ where $r$ and $n$ are an element of the double coset and the size of the coset, respectively. This operation is faster than DoubleCosetsNC (39.9-3) because no double coset objects have to be created.

gap> dc:=DoubleCosetRepsAndSizes(g,u,v);
[ [ (), 12 ], [ (1,3)(2,4), 6 ], [ (1,4)(2,3), 6 ] ]

39.9-6 InfoCoset

39.10 Conjugacy Classes

39.10-1 ConjugacyClass

creates the conjugacy class in G with representative g. This class is an external set, so functions such as Representative (30.4-7) (which returns g), ActingDomain (41.12-3) (which returns G), StabilizerOfExternalSet (41.12-10) (which returns the centralizer of g) and AsList (30.3-8) work for it.

A conjugacy class is an external orbit (see ExternalOrbit (41.12-9)) of group elements with the group acting by conjugation on it. Thus element tests or operation representatives can be computed. The attribute Centralizer (35.4-4) gives the centralizer of the representative (which is the same result as StabilizerOfExternalSet (41.12-10)). (This is a slight abuse of notation: This is not the centralizer of the class as a set which would be the standard behaviour of Centralizer (35.4-4).)

39.10-2 ConjugacyClasses

returns the conjugacy classes of elements of G as a list of class objects of G (see ConjugacyClass (39.10-1) for details). It is guaranteed that the class of the identity is in the first position, the further arrangement depends on the method chosen (and might be different for equal but not identical groups).

For very small groups (of size up to 500) the classes will be computed by the conjugation action of G on itself (see ConjugacyClassesByOrbits (39.10-4)). This can be deliberately switched off using the noaction option shown below.

For solvable groups, the default method to compute the classes is by homomorphic lift (see section 45.17).

ConjugacyClasses supports the following options that can be used to modify this strategy:

gap> g:=SymmetricGroup(4);;
gap> cl:=ConjugacyClasses(g);
[ ()^G, (1,2)^G, (1,2)(3,4)^G, (1,2,3)^G, (1,2,3,4)^G ]
gap> Representative(cl[3]);Centralizer(cl[3]);
(1,2)(3,4)
Group([ (1,2), (1,3)(2,4), (3,4) ])
gap> Size(Centralizer(cl[5]));
4
gap> Size(cl[2]);
6

In general, you will not need to have to influence the method, but simply call ConjugacyClasses –GAP will try to select a suitable method on its own. The method specifications are provided here mainly for expert use.

39.10-3 ConjugacyClassesByRandomSearch

computes the classes of the group G by random search. This works very efficiently for almost simple groups.

This function is also accessible via the option random to the function ConjugacyClass (39.10-1).

39.10-4 ConjugacyClassesByOrbits

computes the classes of the group G as orbits of G on its elements. This can be quick but unsurprisingly may also take a lot of memory if G becomes larger. All the classes will store their element list and thus a membership test will be quick as well.

This function is also accessible via the option action to the function ConjugacyClass (39.10-1).

Typically, for small groups (roughly of order up to $10^3$) the computation of classes as orbits under the action is fastest; memory restrictions (and the increasing cost of eliminating duplicates) make this less efficient for larger groups.

Calculation by random search has the smallest memory requirement, but in generally performs worse, the more classes are there.

The following example shows the effect of this for a small group with many classes:

gap> h:=Group((4,5)(6,7,8),(1,2,3)(5,6,9));;ConjugacyClasses(h:noaction);;time;
110
gap> h:=Group((4,5)(6,7,8),(1,2,3)(5,6,9));;ConjugacyClasses(h:random);;time;
300
gap> h:=Group((4,5)(6,7,8),(1,2,3)(5,6,9));;ConjugacyClasses(h:action);;time;
30

39.10-5 NrConjugacyClasses

gap> g:=Group((1,2,3,4),(1,2));;
gap> NrConjugacyClasses(g);
5

39.10-6 RationalClass

creates the rational class in G with representative g. A rational class consists of all elements that are conjugate to g or to an $i$-th power of g where $i$ is coprime to the order of $g$. Thus a rational class can be interpreted as a conjugacy class of cyclic subgroups. A rational class is an external set (IsExternalSet (41.12-1)) of group elements with the group acting by conjugation on it, but not an external orbit.

39.10-7 RationalClasses

returns a list of the rational classes of the group G. (See RationalClass (39.10-6).)

gap> RationalClasses(DerivedSubgroup(g));
[ RationalClass( AlternatingGroup( [ 1 .. 4 ] ), () ),
  RationalClass( AlternatingGroup( [ 1 .. 4 ] ), (1,2)(3,4) ),
  RationalClass( AlternatingGroup( [ 1 .. 4 ] ), (1,2,3) ) ]

39.10-8 GaloisGroup

Suppose that ratcl is a rational class of a group $G$ with representative $g$. The exponents $i$ for which $g^i$ lies already in the ordinary conjugacy class of $g$, form a subgroup of the prime residue class group $P_n$ (see PrimitiveRootMod (15.3-4)), the so-called Galois group of the rational class. The prime residue class group $P_n$ is obtained in GAP as Units( Integers mod n ), the unit group of a residue class ring. The Galois group of a rational class ratcl is stored in the attribute GaloisGroup as a subgroup of this group.

39.10-9 IsConjugate

tests whether the elements x and y or the subgroups U and V are conjugate under the action of G. (They do not need to be contained in G.) This command is only a shortcut to RepresentativeAction (41.6-1).

gap> IsConjugate(g,Group((1,2,3,4),(1,3)),Group((1,3,2,4),(1,2)));
true

gap> RepresentativeAction(g,(1,2),(3,4));
(1,3)(2,4)

39.10-10 NthRootsInGroup

Let e be an element in the group G. This function returns a list of all those elements in G whose n-th power is e.

gap> NthRootsInGroup(g,(1,2)(3,4),2);
[ (1,3,2,4), (1,4,2,3) ]

39.11 Normal Structure

For the operations Centralizer (35.4-4) and Centre (35.4-5), see Chapter 35.

39.11-1 Normalizer

For two groups G, U, Normalizer computes the normalizer $N_{\textit{G}}(\textit{U})$, that is, the stabilizer of U under the conjugation action of G.

For a group G and a group element g, Normalizer computes $N_{\textit{G}}(\langle \textit{g} \rangle)$.

gap> Normalizer(g,Subgroup(g,[(1,2,3)]));
Group([ (1,2,3), (2,3) ])

39.11-2 Core

If S and U are groups of elements in the same family, this operation returns the core of U in S, that is the intersection of all S-conjugates of U.

gap> g:=Group((1,2,3,4),(1,2));;
gap> Core(g,Subgroup(g,[(1,2,3,4)]));
Group(())

39.11-3 PCore

The p-core of G is the largest normal p-subgroup of G. It is the core of a Sylow p-subgroup of G, see Core (39.11-2).

gap> g:=DicyclicGroup(12);;
gap> PCore(g,2);
Group([ y3 ])
gap> PCore(g,2) = Core(g,SylowSubgroup(g,2));
true
gap> PCore(g,3);
Group([ y*y3 ])
gap> PCore(g,5);
Group([  ])
gap> g:=SymmetricGroup(4);;
gap> PCore(g,2);
Group([ (1,4)(2,3), (1,2)(3,4) ])
gap> PCore(g,2) = Core(g,SylowSubgroup(g,2));
true

39.11-4 NormalClosure

The normal closure of U in G is the smallest normal subgroup of the closure of G and U which contains U.

The second argument may also be a list of group elements, in which case the normal closure of the group generated by these elements is computed.

gap> NormalClosure(g,Subgroup(g,[(1,2,3)])) = Group([ (1,2,3), (2,3,4) ]);
true
gap> NormalClosure(g,[(1,2,3)]) = Group([ (1,2,3), (2,3,4) ]);
true
gap> NormalClosure(g,Group((3,4,5))) = Group([ (3,4,5), (1,5,4), (1,2,5) ]);
true

39.11-5 NormalIntersection

computes the intersection of G and U, assuming that G is normalized by U. This works faster than Intersection, but will not produce the intersection if G is not normalized by U.

gap> NormalIntersection(Group((1,2)(3,4),(1,3)(2,4)),Group((1,2,3,4)));
Group([ (1,3)(2,4) ])

39.11-6 ComplementClassesRepresentatives

Let N be a normal subgroup of G. This command returns a set of representatives for the conjugacy classes of complements of N in G. Complements are subgroups of G which intersect trivially with N and together with N generate G.

At the moment methods are available only for the case that N or G/N is solvable.

gap> ComplementClassesRepresentatives(g,Group((1,2)(3,4),(1,3)(2,4)));
[ Group([ (3,4), (2,4,3) ]) ]

39.11-7 InfoComplement

39.12 Specific and Parametrized Subgroups

The centre of a group (the subgroup of those elements that commute with all other elements of the group) can be computed by the operation Centre (35.4-5).

39.12-1 TrivialSubgroup

gap> TrivialSubgroup(g);
Group(())

39.12-2 CommutatorSubgroup

If G and H are two groups of elements in the same family, this operation returns the group generated by all commutators $[ g, h ] = g^{{-1}} h^{{-1}} g h$ (see Comm (31.12-3)) of elements $g \in \textit{G}$ and $h \in \textit{H}$, that is the group $\left \langle [ g, h ] \mid g \in \textit{G}, h \in \textit{H} \right \rangle$.

gap> CommutatorSubgroup(Group((1,2,3),(1,2)),Group((2,3,4),(3,4)));
Group([ (1,4)(2,3), (1,3,4) ])
gap> Size(last);
12

39.12-3 DerivedSubgroup

The derived subgroup $\textit{G}'$ of G is the subgroup generated by all commutators of pairs of elements of G. It is normal in G and the factor group $\textit{G}/\textit{G}'$ is the largest abelian factor group of G.

gap> g:=Group((1,2,3,4),(1,2));;
gap> DerivedSubgroup(g) = Group([ (1,3,2), (2,4,3) ]);
true

39.12-4 CommutatorLength

returns the minimal number $n$ such that each element in the derived subgroup (see DerivedSubgroup (39.12-3)) of the group G can be written as a product of (at most) $n$ commutators of elements in G.

gap> CommutatorLength( g );
1

39.12-5 FittingSubgroup

gap> FittingSubgroup(g);
Group([ (1,2)(3,4), (1,4)(2,3) ])

39.12-6 FrattiniSubgroup

The Frattini subgroup of a group G is the intersection of all maximal subgroups of G.

gap> FrattiniSubgroup(g);
Group(())

39.12-7 PrefrattiniSubgroup

A factor $M/N$ of G is called a Frattini factor if $M/N$ is contained in the Frattini subgroup of $\textit{G}/N$. A subgroup $P$ is a Prefrattini subgroup of G if $P$ covers each Frattini chief factor of G, and if for each maximal subgroup of G there exists a conjugate maximal subgroup, which contains $P$. In a finite solvable group G the Prefrattini subgroups form a characteristic conjugacy class of subgroups and the intersection of all these subgroups is the Frattini subgroup of G.

gap> G := SmallGroup( 60, 7 );
<pc group of size 60 with 4 generators>
gap> P := PrefrattiniSubgroup(G);
Group([ f2 ])
gap> Size(P);
2
gap> IsNilpotent(P);
true
gap> Core(G,P);
Group([  ])
gap> FrattiniSubgroup(G);
Group([  ])

39.12-8 PerfectResiduum

gap> PerfectResiduum(SymmetricGroup(5));
Alt( [ 1 .. 5 ] )

39.12-9 SolvableRadical

is the solvable radical of the group G, i.e., the largest solvable normal subgroup of G.

gap> rad:= SolvableRadical( SL(2,5) );
<group of 2x2 matrices of size 2 over GF(5)>
gap> Size( rad );
2

39.12-10 Socle

The socle of the group G is the subgroup generated by all minimal normal subgroups.

gap> Socle(g);
Group([ (1,4)(2,3), (1,2)(3,4) ])

39.12-11 SupersolvableResiduum

is the supersolvable residuum of the group G, that is, its smallest normal subgroup $N$ such that the factor group $\textit{G} / N$ is supersolvable.

gap> SupersolvableResiduum(g) = Group([ (1,3)(2,4), (1,4)(2,3) ]);
true

39.12-12 PRump

For a prime $p$, the p-rump of a group G is the subgroup $\textit{G}' \textit{G}^{\textit{p}}$. Unless it equals G itself (which is the e.g. the case if G is perfect), it is equal to the second term of the p-central series of G, see PCentralSeries (39.17-13).

gap> g:=DicyclicGroup(12);;
gap> PRump(g,2) = PCentralSeries(g,2)[2];
true
gap> g:=SymmetricGroup(4);;
gap> PRump(g,2) = AlternatingGroup(4);
true

39.13 Sylow Subgroups and Hall Subgroups

With respect to the following GAP functions, please note that by theorems of P. Hall, a group $G$ is solvable if and only if one of the following conditions holds.

39.13-1 SylowSubgroup

returns a Sylow p-subgroup of the finite group G. This is a p-subgroup of G whose index in G is coprime to p. SylowSubgroup computes Sylow subgroups via the operation SylowSubgroupOp.

gap> g:=SymmetricGroup(4);;
gap> SylowSubgroup(g,2);
Group([ (1,2), (3,4), (1,3)(2,4) ])

39.13-2 SylowComplement

returns a Sylow p-complement of the finite group G. This is a subgroup $U$ of order coprime to p such that the index $[\textit{G}:U]$ is a p-power.

At the moment methods exist only if G is solvable and GAP will issue an error if G is not solvable.

gap> SylowComplement(g,3);
Group([ (1,2), (3,4), (1,3)(2,4) ])

39.13-3 HallSubgroup

computes a P-Hall subgroup for a set P of primes. This is a subgroup the order of which is only divisible by primes in P and whose index is coprime to all primes in P. Such a subgroup is unique up to conjugacy if G is solvable. The function computes Hall subgroups via the operation HallSubgroupOp.

If G is solvable this function always returns a subgroup. If G is not solvable this function might return a subgroup (if it is unique up to conjugacy), a list of subgroups (which are representatives of the conjugacy classes in case there are several such classes) or fail if no such subgroup exists.

gap> h:=SmallGroup(60,10);;
gap> u:=HallSubgroup(h,[2,3]);
Group([ f1, f2, f3 ])
gap> Size(u);
12
gap> h:=PSL(3,5);;
gap> HallSubgroup(h,[2,3]);
[ <permutation group of size 96 with 6 generators>,
  <permutation group of size 96 with 6 generators> ]
gap> u := HallSubgroup(h,[3,31]);;
gap> Size(u); StructureDescription(u);
93
"C31 : C3"
gap> HallSubgroup(h,[5,31]);
fail

39.13-4 SylowSystem

A Sylow system of a group G is a set of Sylow subgroups of G such that every pair of subgroups from this set commutes as subgroups. Sylow systems exist only for solvable groups. The operation returns fail if the group G is not solvable.

gap> h:=SmallGroup(60,10);;
gap> SylowSystem(h);
[ Group([ f1, f2 ]), Group([ f3 ]), Group([ f4 ]) ]
gap> List(last,Size);
[ 4, 3, 5 ]

39.13-5 ComplementSystem

A complement system of a group G is a set of Hall $p'$-subgroups of G, where $p'$ runs through the subsets of prime factors of $|\textit{G}|$ that omit exactly one prime. Every pair of subgroups from this set commutes as subgroups. Complement systems exist only for solvable groups, therefore ComplementSystem returns fail if the group G is not solvable.

gap> ComplementSystem(h);
[ Group([ f3, f4 ]), Group([ f1, f2, f4 ]), Group([ f1, f2, f3 ]) ]
gap> List(last,Size);
[ 15, 20, 12 ]

39.13-6 HallSystem

returns a list containing one Hall $P$-subgroup for each set $P$ of prime divisors of the order of G. Hall systems exist only for solvable groups. The operation returns fail if the group G is not solvable.

gap> HallSystem(h);
[ Group([  ]), Group([ f1, f2 ]), Group([ f1, f2, f3 ]),
  Group([ f1, f2, f3, f4 ]), Group([ f1, f2, f4 ]), Group([ f3 ]),
  Group([ f3, f4 ]), Group([ f4 ]) ]
gap> List(last,Size);
[ 1, 4, 12, 60, 20, 3, 15, 5 ]

39.14 Subgroups characterized by prime powers

39.14-1 Omega

For a p-group G, one defines $\Omega_{\textit{n}}(\textit{G}) = \langle g \in \textit{G} \mid g^{{\textit{p}^{\textit{n}}}} = 1 \rangle$. The default value for n is 1.

gap> h:=SmallGroup(16,10);
<pc group of size 16 with 4 generators>
gap> Omega(h,2);
Group([ f2, f3, f4 ])

39.14-2 Agemo

For a p-group G, one defines $\mho_{\textit{n}}(G) = \langle g^{{\textit{p}^{\textit{n}}}} \mid g \in \textit{G} \rangle$. The default value for n is 1.

gap> Agemo(h,2);Agemo(h,2,2);
Group([ f4 ])
Group([  ])

39.15 Group Properties

Some properties of groups can be defined not only for groups but also for other structures. For example, nilpotency and solvability make sense also for algebras. Note that these names refer to different definitions for groups and algebras, contrary to the situation with finiteness or commutativity. In such cases, the name of the function for groups got a suffix Group to distinguish different meanings for different structures.

Some functions, such as IsPSolvable (39.15-26) and IsPNilpotent (39.15-27), although they are mathematical properties, are not properties in the sense of GAP (see 13.5 and 13.7), as they depend on a parameter.

39.15-1 IsCyclic

A group is cyclic if it can be generated by one element. For a cyclic group, one can compute a generating set consisting of only one element using MinimalGeneratingSet (39.22-3).

39.15-2 IsElementaryAbelian

A group G is elementary abelian if it is commutative and if there is a prime $p$ such that the order of each element in G divides $p$.

39.15-3 IsNilpotentGroup

A group is nilpotent if the lower central series (see LowerCentralSeriesOfGroup (39.17-11) for a definition) reaches the trivial subgroup in a finite number of steps.

39.15-4 NilpotencyClassOfGroup

The nilpotency class of a nilpotent group G is the number of steps in the lower central series of G (see LowerCentralSeriesOfGroup (39.17-11));

39.15-5 IsPerfectGroup

A group is perfect if it equals its derived subgroup (see DerivedSubgroup (39.12-3)).

39.15-6 IsSolvableGroup

A group is solvable if the derived series (see DerivedSeriesOfGroup (39.17-7) for a definition) reaches the trivial subgroup in a finite number of steps.

For finite groups this is the same as being polycyclic (see IsPolycyclicGroup (39.15-7)), and each polycyclic group is solvable, but there are infinite solvable groups that are not polycyclic.

39.15-7 IsPolycyclicGroup

A group is polycyclic if it has a subnormal series with cyclic factors. For finite groups this is the same as if the group is solvable (see IsSolvableGroup (39.15-6)).

39.15-8 IsSupersolvableGroup

A finite group is supersolvable if it has a normal series with cyclic factors.

39.15-9 IsMonomialGroup

A finite group is monomial if every irreducible complex character is induced from a linear character of a subgroup.

39.15-10 IsSimpleGroup

A group is simple if it is nontrivial and has no nontrivial normal subgroups. A nonabelian simple group is simple and not abelian.

39.15-11 IsAlmostSimpleGroup

A group G is almost simple if a nonabelian simple group $S$ exists such that G is isomorphic to a subgroup of the automorphism group of $S$ that contains all inner automorphisms of $S$.

Equivalently, G is almost simple if and only if it has a unique minimal normal subgroup $N$ and if $N$ is a nonabelian simple group.

Note that an almost simple group is not defined as an extension of a simple group by outer automorphisms, since we want to exclude extensions of groups of prime order. In particular, a simple group is almost simple if and only if it is nonabelian.

gap> IsAlmostSimpleGroup( AlternatingGroup( 5 ) );
true
gap> IsAlmostSimpleGroup( SymmetricGroup( 5 ) );
true
gap> IsAlmostSimpleGroup( SymmetricGroup( 3 ) );
false
gap> IsAlmostSimpleGroup( SL( 2, 5 ) );
false

39.15-12 IsQuasisimpleGroup

A group G is quasisimple if G is perfect (see IsPerfectGroup (39.15-5)) and if G$/Z($G$)$ is simple (see IsSimpleGroup (39.15-10)), where $Z($G$)$ is the centre of G (see Centre (35.4-5)).

gap> IsQuasisimpleGroup( AlternatingGroup( 5 ) );
true
gap> IsQuasisimpleGroup( SymmetricGroup( 5 ) );
false
gap> IsQuasisimpleGroup( SL( 2, 5 ) );
true

39.15-13 IsomorphismTypeInfoFiniteSimpleGroup

For a finite simple group G, IsomorphismTypeInfoFiniteSimpleGroup returns a record with the components name, shortname, series, and possibly parameter, describing the isomorphism type of G.

The values of the components name, shortname, and series are strings, name gives name(s) for G, shortname gives one name for G that is compatible with the naming scheme used in the GAP packages CTblLib and AtlasRep (and in the Atlas of Finite Groups [CCN+85]), and series describes the following series.

(If different characterizations of G are possible only one is given by series and parameter, while name may give several names.)

An equal sign in the name denotes different naming schemes for the same group, a tilde sign abstract isomorphisms between groups constructed in a different way.

gap> IsomorphismTypeInfoFiniteSimpleGroup(
>                             Group((4,5)(6,7),(1,2,4)(3,5,6)));
rec(
  name := "A(1,7) = L(2,7) ~ B(1,7) = O(3,7) ~ C(1,7) = S(2,7) ~ 2A(1,\
7) = U(2,7) ~ A(2,2) = L(3,2)", parameter := [ 2, 7 ], series := "L",
  shortname := "L3(2)" )

For a positive integer n, IsomorphismTypeInfoFiniteSimpleGroup returns fail if n is not the order of a finite simple group, and a record as described for the case of a group G otherwise. If more than one simple group of order n exists then the result record contains only the name component, a string that lists the two possible isomorphism types of simple groups of this order.

gap> IsomorphismTypeInfoFiniteSimpleGroup( 5 );
rec( name := "Z(5)", parameter := 5, series := "Z", shortname := "C5"
 )
gap> IsomorphismTypeInfoFiniteSimpleGroup( 6 );
fail
gap> IsomorphismTypeInfoFiniteSimpleGroup(Size(SymplecticGroup(6,3))/2);
rec(
  name := "cannot decide from size alone between B(3,3) = O(7,3) and C\
(3,3) = S(6,3)", parameter := [ 3, 3 ] )

39.15-14 SimpleGroup

This function will construct an instance of the specified nonabelian simple group. Groups are specified via their name in ATLAS style notation, with parameters added if necessary. The intelligence applied to parsing the name is limited, and at the moment no proper extensions can be constructed. For groups which do not have a permutation representation of small degree the AtlasRep package might need to be installed to construct these groups.

gap> g:=SimpleGroup("M(23)");
M23
gap> Size(g);
10200960
gap> g:=SimpleGroup("PSL",3,5);
PSL(3,5)
gap> Size(g);
372000
gap> g:=SimpleGroup("PSp6",2);
PSp(6,2)

39.15-15 SimpleGroupsIterator

This function returns an iterator that will run over all nonabelian simple groups, starting at order start if specified, up to order $10^{27}$ (or -- if specified -- order end). If the option NOPSL2 is given, groups of type $PSL_2(q)$ are omitted.

gap> it:=SimpleGroupsIterator(20000);
<iterator>
gap> List([1..8],x->NextIterator(it));
[ A8, PSL(3,4), PSL(2,37), PSp(4,3), Sz(8), PSL(2,32), PSL(2,41),
  PSL(2,43) ]
gap> it:=SimpleGroupsIterator(1,2000);;
gap> l:=[];;for i in it do Add(l,i);od;l;
[ A5, PSL(2,7), A6, PSL(2,8), PSL(2,11), PSL(2,13) ]
gap> it:=SimpleGroupsIterator(20000,100000:NOPSL2);;
gap> l:=[];;for i in it do Add(l,i);od;l;
[ A8, PSL(3,4), PSp(4,3), Sz(8), PSU(3,4), M12 ]

39.15-16 SmallSimpleGroup

Returns: The ith simple group of order order in the stored list, given in a small-degree permutation representation, or fail (20.2-1) if no such simple group exists.

If i is not given, it defaults to 1. Currently, all simple groups of order less than $10^6$ are available via this function.

gap> SmallSimpleGroup(60);
A5
gap> SmallSimpleGroup(20160,1);
A8
gap> SmallSimpleGroup(20160,2);
PSL(3,4)

39.15-17 AllSmallNonabelianSimpleGroups

Returns: A list of all nonabelian simple groups whose order lies in the range orders.

The groups are given in small-degree permutation representations. The returned list is sorted by ascending group order. Currently, all simple groups of order less than $10^6$ are available via this function.

gap> List(AllSmallNonabelianSimpleGroups([1..1000000]),
>         StructureDescription);
[ "A5", "PSL(3,2)", "A6", "PSL(2,8)", "PSL(2,11)", "PSL(2,13)",
  "PSL(2,17)", "A7", "PSL(2,19)", "PSL(2,16)", "PSL(3,3)",
  "PSU(3,3)", "PSL(2,23)", "PSL(2,25)", "M11", "PSL(2,27)",
  "PSL(2,29)", "PSL(2,31)", "A8", "PSL(3,4)", "PSL(2,37)", "O(5,3)",
  "Sz(8)", "PSL(2,32)", "PSL(2,41)", "PSL(2,43)", "PSL(2,47)",
  "PSL(2,49)", "PSU(3,4)", "PSL(2,53)", "M12", "PSL(2,59)",
  "PSL(2,61)", "PSU(3,5)", "PSL(2,67)", "J1", "PSL(2,71)", "A9",
  "PSL(2,73)", "PSL(2,79)", "PSL(2,64)", "PSL(2,81)", "PSL(2,83)",
  "PSL(2,89)", "PSL(3,5)", "M22", "PSL(2,97)", "PSL(2,101)",
  "PSL(2,103)", "HJ", "PSL(2,107)", "PSL(2,109)", "PSL(2,113)",
  "PSL(2,121)", "PSL(2,125)", "O(5,4)" ]

39.15-18 IsFinitelyGeneratedGroup

tests whether the group G can be generated by a finite number of generators. (This property is mainly used to obtain finiteness conditions.)

Note that this is a pure existence statement. Even if a group is known to be generated by a finite number of elements, it can be very hard or even impossible to obtain such a generating set if it is not known.

39.15-19 IsSubsetLocallyFiniteGroup

A group is called locally finite if every finitely generated subgroup is finite. This property checks whether the group U is a subset of a locally finite group. This is used to check whether finite generation will imply finiteness, as it does for example for permutation groups.

39.15-20 IsPGroup

A $p$-group is a group in which the order (see Order (31.10-10)) of every element is of the form $p^n$ for a prime integer $p$ and a nonnegative integer $n$. IsPGroup returns true if G is a $p$-group, and false otherwise.

Finite $p$-groups are precisely those groups whose order (see Size (30.4-6)) is $1$ or a prime power (see IsPrimePowerInt (14.4-4), and are always nilpotent.

Note that $p$-groups can also be infinite, and in that case, need not be nilpotent.

gap> IsPGroup( DihedralGroup( 8 ) );
true
gap> IsPGroup( TrivialGroup() );
true
gap> IsPGroup( DihedralGroup( 10 ) );
false

39.15-21 IsPowerfulPGroup

A finite $p$-group G is said to be a powerful $p$-group if the commutator subgroup $[\textit{G},\textit{G}]$ is contained in $\textit{G}^{p}$ if the prime $p$ is odd, or if $[\textit{G},\textit{G}]$ is contained in $\textit{G}^{4}$ if $p = 2$. The subgroup $\textit{G}^{p}$ is called the first Agemo subgroup, (see Agemo (39.14-2)). IsPowerfulPGroup returns true if G is a powerful $p$-group, and false otherwise. Note: This function returns true if G is the trivial group.

39.15-22 IsRegularPGroup

A finite $p$-group G is a regular $p$-group if for all $a,b$ in G, one has $a^p b^p = (a b)^p c^p$ where $c$ is an element of the derived subgroup of the group generated by $a$ and $b$ (see [Hal34]). IsRegularPGroup returns true if G is a regular $p$-group, and false otherwise. Note: This function returns true if G is the trivial group.

39.15-23 PrimePGroup

If G is a nontrivial $p$-group (see IsPGroup (39.15-20)), PrimePGroup returns the prime integer $p$; if G is trivial then PrimePGroup returns fail. Otherwise an error is issued.

(One should avoid a common error of writing if IsPGroup(g) then ... PrimePGroup(g) ... where the code represented by dots assumes that PrimePGroup(g) is an integer.)

39.15-24 PClassPGroup

The $p$-class of a $p$-group G (see IsPGroup (39.15-20)) is the length of the lower $p$-central series (see PCentralSeries (39.17-13)) of G. If G is not a $p$-group then an error is issued.

39.15-25 RankPGroup

For a $p$-group G (see IsPGroup (39.15-20)), RankPGroup returns the rank of G, which is defined as the minimal size of a generating system of G. If G is not a $p$-group then an error is issued.

gap> h:=Group((1,2,3,4),(1,3));;
gap> PClassPGroup(h);
2
gap> RankPGroup(h);
2

39.15-26 IsPSolvable

A finite group is $p$-solvable if every chief factor either has order not divisible by $p$, or is solvable.

39.15-27 IsPNilpotent

39.16 Numerical Group Attributes

This section gives only some examples of numerical group attributes, so it should not serve as a collection of all numerical group attributes. The manual contains more such attributes documented in this manual, for example, NrConjugacyClasses (39.10-5), NilpotencyClassOfGroup (39.15-4) and others.

Note also that some functions, such as EulerianFunction (39.16-3), are mathematical attributes, but not GAP attributes (see 13.5) as they are depending on a parameter.

39.16-1 AbelianInvariants

returns the abelian invariants (also sometimes called primary decomposition) of the commutator factor group of the group G. These are given as a list of prime-powers or zeroes and describe the structure of $\textit{G}/\textit{G}'$ as a direct product of cyclic groups of prime power (or infinite) order.

(See IndependentGeneratorsOfAbelianGroup (39.22-5) to obtain actual generators).

gap> g:=Group((1,2,3,4),(1,2),(5,6));;
gap> AbelianInvariants(g);
[ 2, 2 ]
gap> h:=FreeGroup(2);;h:=h/[h.1^3];;
gap> AbelianInvariants(h);
[ 0, 3 ]

39.16-2 Exponent

The exponent $e$ of a group G is the lcm of the orders of its elements, that is, $e$ is the smallest integer such that $g^e = 1$ for all $g \in \textit{G}$.

gap> Exponent(g);
12

39.16-3 EulerianFunction

returns the number of n-tuples $(g_1, g_2, \ldots, g_n)$ of elements of the group G that generate the whole group G. The elements of such an n-tuple need not be different.

In [Hal36], the notation $\phi_{\textit{n}}(\textit{G})$ is used for the value returned by EulerianFunction, and the quotient of $\phi_{\textit{n}}(\textit{G})$ by the order of the automorphism group of G is called $d_{\textit{n}}(\textit{G})$. If G is a nonabelian simple group then $d_{\textit{n}}(\textit{G})$ is the greatest number $d$ for which the direct product of $d$ groups isomorphic with G can be generated by n elements.

If the Library of Tables of Marks (see Chapter 70) covers the group G, you may also use EulerianFunctionByTom (70.9-9).

gap> EulerianFunction( g, 2 );
432

39.17 Subgroup Series

In group theory many subgroup series are considered, and GAP provides commands to compute them. In the following sections, there is always a series $G = U_1 > U_2 > \cdots > U_m = \langle 1 \rangle$ of subgroups considered. A series also may stop without reaching $G$ or $\langle 1 \rangle$.

A series of normal subgroups is called central if $U_i/U_{{i+1}}$ is central in $G / U_{{i+1}}$.

We call a series refinable if intermediate subgroups can be added to the series without destroying the properties of the series.

Unless explicitly declared otherwise, all subgroup series are descending. That is they are stored in decreasing order.

39.17-1 ChiefSeries

is a series of normal subgroups of G which cannot be refined further. That is there is no normal subgroup $N$ of G with $U_i > N > U_{{i+1}}$. This attribute returns one chief series (of potentially many possibilities).

gap> g:=Group((1,2,3,4),(1,2));;
gap> ChiefSeries(g);
[ Group([ (1,2,3,4), (1,2) ]),
  Group([ (2,4,3), (1,4)(2,3), (1,3)(2,4) ]),
  Group([ (1,4)(2,3), (1,3)(2,4) ]), Group(()) ]

39.17-2 ChiefSeriesThrough

is a chief series of the group G going through the normal subgroups in the list l, which must be a list of normal subgroups of G contained in each other, sorted by descending size. This attribute returns one chief series (of potentially many possibilities).

39.17-3 ChiefSeriesUnderAction

returns a series of normal subgroups of G which are invariant under H such that the series cannot be refined any further. G must be a subgroup of H. This attribute returns one such series (of potentially many possibilities).

39.17-4 SubnormalSeries

If U is a subgroup of G this operation returns a subnormal series that descends from G to a subnormal subgroup $V \geq $U. If U is subnormal, $V =$ U.

gap> s:=SubnormalSeries(g,Group((1,2)(3,4))) =
> [ Group([ (1,2,3,4), (1,2) ]),
>   Group([ (1,2)(3,4), (1,3)(2,4) ]),
>   Group([ (1,2)(3,4) ]) ];
true

39.17-5 CompositionSeries

A composition series is a subnormal series which cannot be refined. This attribute returns one composition series (of potentially many possibilities). The variant CompositionSeriesThrough takes as second argument a list normals of normal subgroups of the group, and returns a composition series that incorporates these normal subgroups.

39.17-6 DisplayCompositionSeries

Displays a composition series of G in a nice way, identifying the simple factors.

gap> CompositionSeries(g);
[ Group([ (3,4), (2,4,3), (1,4)(2,3), (1,3)(2,4) ]),
  Group([ (2,4,3), (1,4)(2,3), (1,3)(2,4) ]),
  Group([ (1,4)(2,3), (1,3)(2,4) ]), Group([ (1,3)(2,4) ]), Group(())
 ]
gap> DisplayCompositionSeries(Group((1,2,3,4,5,6,7),(1,2)));
G (2 gens, size 5040)
 | C2
S (5 gens, size 2520)
 | A7
1 (0 gens, size 1)

39.17-7 DerivedSeriesOfGroup

The derived series of a group is obtained by $U_{{i+1}} = U_i'$. It stops if $U_i$ is perfect.

39.17-8 DerivedLength

The derived length of a group is the number of steps in the derived series. (As there is always the group, it is the series length minus 1.)

gap> List(DerivedSeriesOfGroup(g),Size);
[ 24, 12, 4, 1 ]
gap> DerivedLength(g);
3

39.17-9 ElementaryAbelianSeries

returns a series of normal subgroups of $G$ such that all factors are elementary abelian. If the group is not solvable (and thus no such series exists) it returns fail.

The variant ElementaryAbelianSeriesLargeSteps tries to make the steps in this series large (by eliminating intermediate subgroups if possible) at a small additional cost.

In the third variant, an elementary abelian series through the given series of normal subgroups in the list list is constructed.

gap> List(ElementaryAbelianSeries(g),Size);
[ 24, 12, 4, 1 ]

39.17-10 InvariantElementaryAbelianSeries

For a (solvable) group G and a list of automorphisms morph of G, this command finds a normal series of G with elementary abelian factors such that every group in this series is invariant under every automorphism in morph.

If a normal subgroup N of G which is invariant under morph is given, this series is chosen to contain N. No tests are performed to check the validity of the arguments.

The series obtained will be constructed to prefer large steps unless fine is given as true.

gap> g:=Group((1,2,3,4),(1,3));
Group([ (1,2,3,4), (1,3) ])
gap> hom:=GroupHomomorphismByImages(g,g,GeneratorsOfGroup(g),
> [(1,4,3,2),(1,4)(2,3)]);
[ (1,2,3,4), (1,3) ] -> [ (1,4,3,2), (1,4)(2,3) ]
gap> InvariantElementaryAbelianSeries(g,[hom]);
[ Group([ (1,2,3,4), (1,3) ]), Group([ (1,3)(2,4) ]), Group(()) ]

39.17-11 LowerCentralSeriesOfGroup

The lower central series of a group G is defined as $U_{{i+1}}:= [\textit{G}, U_i]$. It is a central series of normal subgroups. The name derives from the fact that $U_i$ is contained in the $i$-th step subgroup of any central series.

39.17-12 UpperCentralSeriesOfGroup

The upper central series of a group G is defined as an ending series $U_i / U_{{i+1}}:= Z(\textit{G}/U_{{i+1}})$. It is a central series of normal subgroups. The name derives from the fact that $U_i$ contains every $i$-th step subgroup of a central series.

39.17-13 PCentralSeries

The p-central series of G is defined by $U_1:= \textit{G}$, $U_i:= [\textit{G}, U_{{i-1}}] U_{{i-1}}^{\textit{p}}$.

gap> g:=DicyclicGroup(12);;
gap> PCentralSeries(g,2);
[ <pc group of size 12 with 3 generators>, Group([ y3, y*y3 ]), Group([ y*y3 ]) ]
gap> g:=SymmetricGroup(4);;
gap> List(PCentralSeries(g,2), StructureDescription);
[ "S4", "A4" ]

39.17-14 JenningsSeries

For a $p$-group G, this function returns its Jennings series. This series is defined by setting $G_1 = \textit{G}$ and for $i \geq 0$, $G_{{i+1}} = [G_i,\textit{G}] G_j^p$, where $j$ is the smallest integer $> i/p$.

39.17-15 DimensionsLoewyFactors

This operation computes the dimensions of the factors of the Loewy series of G. (See [HB82, p. 157] for the slightly complicated definition of the Loewy Series.)

The dimensions are computed via the JenningsSeries (39.17-14) without computing the Loewy series itself.

gap> G:= SmallGroup( 3^6, 100 );
<pc group of size 729 with 6 generators>
gap> JenningsSeries( G );
[ <pc group of size 729 with 6 generators>, Group([ f3, f4, f5, f6 ]),
  Group([ f4, f5, f6 ]), Group([ f5, f6 ]), Group([ f5, f6 ]),
  Group([ f5, f6 ]), Group([ f6 ]), Group([ f6 ]), Group([ f6 ]),
  Group([ <identity> of ... ]) ]
gap> DimensionsLoewyFactors(G);
[ 1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26,
  27, 27, 27, 27, 27, 27, 27, 27, 27, 26, 25, 23, 22, 20, 19, 17, 16,
  14, 13, 11, 10, 8, 7, 5, 4, 2, 1 ]

39.17-16 AscendingChain

This function computes an ascending chain of subgroups from U to G. This chain is given as a list whose first entry is U and the last entry is G. The function tries to make the links in this chain small.

The option refineIndex can be used to give a bound for refinements of steps to avoid GAP trying to enforce too small steps. The option cheap (if set to true) will overall limit the amount of heuristic searches.

39.17-17 IntermediateGroup

This routine tries to find a subgroup $E$ of G, such that $\textit{G} > E > \textit{U}$ holds. If U is maximal in G, the function returns fail. This is done by finding minimal blocks for the operation of G on the right cosets of U.

39.17-18 IntermediateSubgroups

returns a list of all subgroups of G that properly contain U; that is all subgroups between G and U. It returns a record with a component subgroups, which is a list of these subgroups, as well as a component inclusions, which lists all maximality inclusions among these subgroups. A maximality inclusion is given as a list $[i, j]$ indicating that the subgroup number $i$ is a maximal subgroup of the subgroup number $j$, the numbers $0$ and $1 +$ Length(subgroups) are used to denote U and G, respectively.

39.17-19 StructuralSeriesOfGroup

The structural series of a finite group G is a descending series of characteristic subgroups which goes through the derived series of the solvable radical of $G$, refined into elementary abelian factors, as well as the socle and the Pker (kernel of the action on socle components) of the radical factor

gap> gp:=WreathProduct(SymmetricGroup(5),SymmetricGroup(3));;
gap> gp:=WreathProduct(Group((1,2,3,4,5,6)),gp);;
gap> List(StructuralSeriesOfGroup(gp),Size);
[ 4874877920083968000, 812479653347328000, 101559956668416000, 470184984576, 32768, 1 ]

39.18 Factor Groups

39.18-1 NaturalHomomorphismByNormalSubgroup

returns a homomorphism from G to another group whose kernel is N. GAP will try to select the image group as to make computations in it as efficient as possible. As the factor group $\textit{G}/\textit{N}$ can be identified with the image of G this permits efficient computations in the factor group. The homomorphism returned is not necessarily surjective, so ImagesSource (32.4-1) should be used instead of Range (32.3-7) to get a group isomorphic to the factor group. The NC variant does not check whether N is normal in G.

39.18-2 FactorGroup

returns the image of the NaturalHomomorphismByNormalSubgroup(G,N). This function is provided for compatibility with older code, but if a connection between group and factor is desired, users need to start by obtaining the NaturalHomomorphismByNormalSubgroup in the first place. The NC version does not test whether N is normal in G.

gap> g:=Group((1,2,3,4),(1,2));;n:=Subgroup(g,[(1,2)(3,4),(1,3)(2,4)]);;
gap> hom:=NaturalHomomorphismByNormalSubgroup(g,n);
[ (1,2,3,4), (1,2) ] -> [ f1*f2, f1 ]
gap> Size(ImagesSource(hom));
6
gap> StructureDescription(Image(hom,g));
"S3"

39.18-3 CommutatorFactorGroup

computes the commutator factor group $\textit{G}/\textit{G}'$ of the group G.

gap> CommutatorFactorGroup(g);
Group([ f1 ])

39.18-4 MaximalAbelianQuotient

returns an epimorphism from G onto the maximal abelian quotient of G. The kernel of this epimorphism is the derived subgroup of G, see DerivedSubgroup (39.12-3).

39.18-5 HasAbelianFactorGroup

tests whether G $/$ N is abelian (without explicitly constructing the factor group and without testing whether N is in fact a normal subgroup).

gap> HasAbelianFactorGroup(g,n);
false
gap> HasAbelianFactorGroup(DerivedSubgroup(g),n);
true

39.18-6 HasElementaryAbelianFactorGroup

tests whether G $/$ N is elementary abelian (without explicitly constructing the factor group and without testing whether N is in fact a normal subgroup).

39.18-7 CentralizerModulo

Computes the full preimage of the centralizer $C_{{\textit{G}/\textit{N}}}(\textit{elm} \cdot \textit{N})$ in G (without necessarily constructing the factor group).

gap> CentralizerModulo(g,n,(1,2));
Group([ (3,4), (1,3)(2,4), (1,4)(2,3) ])

39.19 Sets of Subgroups

39.19-1 ConjugacyClassSubgroups

generates the conjugacy class of subgroups of G with representative U. This class is an external set, so functions such as Representative (30.4-7), (which returns U), ActingDomain (41.12-3) (which returns G), StabilizerOfExternalSet (41.12-10) (which returns the normalizer of U), and AsList (30.3-8) work for it.

(The use of the [] list access to select elements of the class is considered obsolescent and will be removed in future versions. Use ClassElementLattice (39.20-2) instead.)

gap> g:=Group((1,2,3,4),(1,2));;IsNaturalSymmetricGroup(g);;
gap> cl:=ConjugacyClassSubgroups(g,Subgroup(g,[(1,2)]));
Group( [ (1,2) ] )^G
gap> Size(cl);
6
gap> ClassElementLattice(cl,4);
Group([ (2,3) ])

39.19-2 IsConjugacyClassSubgroupsRep

Is the representation GAP uses for conjugacy classes of subgroups. It can be used to check whether an object is a class of subgroups. The second representation IsConjugacyClassSubgroupsByStabilizerRep in addition is an external orbit by stabilizer and will compute its elements via a transversal of the stabilizer.

39.19-3 ConjugacyClassesSubgroups

This attribute returns a list of all conjugacy classes of subgroups of the group G. It also is applicable for lattices of subgroups (see LatticeSubgroups (39.20-1)). The order in which the classes are listed depends on the method chosen by GAP. For each class of subgroups, a representative can be accessed using Representative (30.4-7).

gap> ConjugacyClassesSubgroups(g);
[ Group( () )^G, Group( [ (1,3)(2,4) ] )^G, Group( [ (3,4) ] )^G,
  Group( [ (2,4,3) ] )^G, Group( [ (1,4)(2,3), (1,3)(2,4) ] )^G,
  Group( [ (3,4), (1,2)(3,4) ] )^G,
  Group( [ (1,3,2,4), (1,2)(3,4) ] )^G, Group( [ (3,4), (2,4,3) ] )^G,
  Group( [ (1,4)(2,3), (1,3)(2,4), (3,4) ] )^G,
  Group( [ (1,4)(2,3), (1,3)(2,4), (2,4,3) ] )^G,
  Group( [ (1,4)(2,3), (1,3)(2,4), (2,4,3), (3,4) ] )^G ]

39.19-4 ConjugacyClassesMaximalSubgroups

returns the conjugacy classes of maximal subgroups of G. Representatives of the classes can be computed directly by MaximalSubgroupClassReps (39.19-5).

gap> ConjugacyClassesMaximalSubgroups(g);
[ Group( [ (2,4,3), (1,4)(2,3), (1,3)(2,4) ] )^G,
  Group( [ (3,4), (1,3)(2,4) ] )^G, Group( [ (3,4), (2,4,3) ] )^G ]

39.19-5 MaximalSubgroupClassReps

gap> MaximalSubgroupClassReps(g);
[ Group([ (2,4,3), (1,4)(2,3), (1,3)(2,4) ]),
  Group([ (3,4), (1,3)(2,4) ]), Group([ (3,4), (2,4,3) ]) ]

39.19-6 LowIndexSubgroups

The operation LowIndexSubgroups computes representatives of the conjugacy classes of subgroups of the group G that have index less than or equal to index.

For finitely presented groups this operation simply defaults to LowIndexSubgroupsFpGroup (47.10-1). In other cases, it uses repeated calculation of maximal subgroups.

The function LowLayerSubgroups (39.20-6) works similar but does not bound the index, but instead considers up to layer-th maximal subgroups.

gap> g:=TransitiveGroup(18,950);;
gap> l:=LowIndexSubgroups(g,20);;Collected(List(l,x->Index(g,x)));
[ [ 1, 1 ], [ 2, 1 ], [ 5, 1 ], [ 6, 1 ], [ 10, 2 ], [ 12, 3 ], [ 15, 1 ],
  [ 16, 2 ], [ 18, 1 ], [ 20, 9 ] ]

39.19-7 AllSubgroups

For a finite group G AllSubgroups returns a list of all subgroups of G, intended primarily for use in class for small examples. This list will quickly get very long and in general use of ConjugacyClassesSubgroups (39.19-3) is recommended.

gap> AllSubgroups(SymmetricGroup(3));
[ Group(()), Group([ (2,3) ]), Group([ (1,2) ]), Group([ (1,3) ]),
  Group([ (1,2,3) ]), Group([ (1,2,3), (2,3) ]) ]

39.19-8 MaximalSubgroups

returns a list of all maximal subgroups of G. This may take up much space, therefore the command should be avoided if possible. See ConjugacyClassesMaximalSubgroups (39.19-4).

gap> MaximalSubgroups(Group((1,2,3),(1,2)));
[ Group([ (1,2,3) ]), Group([ (2,3) ]), Group([ (1,2) ]),
  Group([ (1,3) ]) ]

39.19-9 NormalSubgroups

gap> g:=SymmetricGroup(4);;
gap> List( NormalSubgroups(g), StructureDescription );
[ "S4", "A4", "C2 x C2", "1" ]
gap> g:=AbelianGroup([2,2]);; NormalSubgroups(g);
[ <pc group of size 4 with 2 generators>, Group([ f2 ]),
  Group([ f1*f2 ]), Group([ f1 ]), Group([  ]) ]

39.19-10 MaximalNormalSubgroups

is a list containing those proper normal subgroups of the group G that are maximal among the proper normal subgroups. Gives error if G/G' is infinite, yielding infinitely many maximal normal subgroups. Note, that the maximal normal subgroups of a group G can be computed more efficiently if the character table of G is known or if G is known to be abelian or solvable (even if infinite). So if the character table is needed, anyhow, or G is suspected to be abelian or solvable, then these should be computed before computing the maximal normal subgroups.

gap> g:=SymmetricGroup(4);; MaximalNormalSubgroups( g );
[ Alt( [ 1 .. 4 ] ) ]
gap> f := FreeGroup("x", "y");; x := f.1;; y := f.2;;
gap> List(MaximalNormalSubgroups(f/[x^2, y^2]), GeneratorsOfGroup);
[ [ x, y*x*y^-1 ], [ y, x*y*x^-1 ], [ y*x^-1 ] ]

39.19-11 MinimalNormalSubgroups

is a list containing those nontrivial normal subgroups of the group G that are minimal among the nontrivial normal subgroups.

gap> g:=SymmetricGroup(4);; MinimalNormalSubgroups( g );
[ Group([ (1,4)(2,3), (1,3)(2,4) ]) ]

39.19-12 CharacteristicSubgroups

returns a list of all characteristic subgroups of G, that is subgroups that are invariant under all automorphisms.

gap> g:=SymmetricGroup(4);;
gap> List( CharacteristicSubgroups(g), StructureDescription );
[ "S4", "A4", "C2 x C2", "1" ]
gap> g:=AbelianGroup([2,2]);; CharacteristicSubgroups(g);
[ <pc group of size 4 with 2 generators>, Group([  ]) ]

39.20 Subgroup Lattice

39.20-1 LatticeSubgroups

computes the lattice of subgroups of the group G. This lattice has the conjugacy classes of subgroups as attribute ConjugacyClassesSubgroups (39.19-3) and permits one to test maximality/minimality relations.

gap> g:=SymmetricGroup(4);;
gap> l:=LatticeSubgroups(g);
<subgroup lattice of Sym( [ 1 .. 4 ] ), 11 classes, 30 subgroups>
gap> ConjugacyClassesSubgroups(l);
[ Group( () )^G, Group( [ (1,3)(2,4) ] )^G, Group( [ (3,4) ] )^G,
  Group( [ (2,4,3) ] )^G, Group( [ (1,4)(2,3), (1,3)(2,4) ] )^G,
  Group( [ (3,4), (1,2)(3,4) ] )^G,
  Group( [ (1,3,2,4), (1,2)(3,4) ] )^G, Group( [ (3,4), (2,4,3) ] )^G,
  Group( [ (1,4)(2,3), (1,3)(2,4), (3,4) ] )^G,
  Group( [ (1,4)(2,3), (1,3)(2,4), (2,4,3) ] )^G,
  Group( [ (1,4)(2,3), (1,3)(2,4), (2,4,3), (3,4) ] )^G ]

39.20-2 ClassElementLattice

For a class C of subgroups, obtained by a lattice computation, this operation returns the n-th conjugate subgroup in the class.

Because of other methods installed, calling AsList (30.3-8) with C can give a different arrangement of the class elements!

The GAP package XGAP permits a graphical display of the lattice of subgroups in a nice way.

39.20-3 DotFileLatticeSubgroups

This function produces a graphical representation of the subgroup lattice L in file file. The output is in .dot (also known as GraphViz format). For details on the format, and information about how to display or edit this format see https://www.graphviz.org. (On the Macintosh, the program OmniGraffle is also able to read this format.)

Subgroups are labelled in the form i-j where i is the number of the class of subgroups and j the number within this class. Normal subgroups are represented by a box.

gap> DotFileLatticeSubgroups(l,"s4lat.dot");

39.20-4 MaximalSubgroupsLattice

For a lattice lat of subgroups this attribute contains the maximal subgroup relations among the subgroups of the lattice. It is a list corresponding to the ConjugacyClassesSubgroups (39.19-3) value of the lattice, each entry giving a list of the maximal subgroups of the representative of this class. Every maximal subgroup is indicated by a list of the form $[ c, n ]$ which means that the $n$-th subgroup in class number $c$ is a maximal subgroup of the representative.

The number $n$ corresponds to access via ClassElementLattice (39.20-2) and not necessarily the AsList (30.3-8) arrangement! See also MinimalSupergroupsLattice (39.20-5).

gap> MaximalSubgroupsLattice(l);
[ [  ], [ [ 1, 1 ] ], [ [ 1, 1 ] ], [ [ 1, 1 ] ],
  [ [ 2, 1 ], [ 2, 2 ], [ 2, 3 ] ], [ [ 3, 1 ], [ 3, 6 ], [ 2, 3 ] ],
  [ [ 2, 3 ] ], [ [ 4, 1 ], [ 3, 1 ], [ 3, 2 ], [ 3, 3 ] ],
  [ [ 7, 1 ], [ 6, 1 ], [ 5, 1 ] ],
  [ [ 5, 1 ], [ 4, 1 ], [ 4, 2 ], [ 4, 3 ], [ 4, 4 ] ],
  [ [ 10, 1 ], [ 9, 1 ], [ 9, 2 ], [ 9, 3 ], [ 8, 1 ], [ 8, 2 ],
      [ 8, 3 ], [ 8, 4 ] ] ]
gap> last[6];
[ [ 3, 1 ], [ 3, 6 ], [ 2, 3 ] ]
gap> u1:=Representative(ConjugacyClassesSubgroups(l)[6]);
Group([ (3,4), (1,2)(3,4) ])
gap> u2:=ClassElementLattice(ConjugacyClassesSubgroups(l)[3],1);;
gap> u3:=ClassElementLattice(ConjugacyClassesSubgroups(l)[3],6);;
gap> u4:=ClassElementLattice(ConjugacyClassesSubgroups(l)[2],3);;
gap> IsSubgroup(u1,u2);IsSubgroup(u1,u3);IsSubgroup(u1,u4);
true
true
true

39.20-5 MinimalSupergroupsLattice

For a lattice lat of subgroups this attribute contains the minimal supergroup relations among the subgroups of the lattice. It is a list corresponding to the ConjugacyClassesSubgroups (39.19-3) value of the lattice, each entry giving a list of the minimal supergroups of the representative of this class. Every minimal supergroup is indicated by a list of the form $[ c, n ]$, which means that the $n$-th subgroup in class number $c$ is a minimal supergroup of the representative.

The number $n$ corresponds to access via ClassElementLattice (39.20-2) and not necessarily the AsList (30.3-8) arrangement! See also MaximalSubgroupsLattice (39.20-4).

gap> MinimalSupergroupsLattice(l);
[ [ [ 2, 1 ], [ 2, 2 ], [ 2, 3 ], [ 3, 1 ], [ 3, 2 ], [ 3, 3 ],
      [ 3, 4 ], [ 3, 5 ], [ 3, 6 ], [ 4, 1 ], [ 4, 2 ], [ 4, 3 ],
      [ 4, 4 ] ], [ [ 5, 1 ], [ 6, 2 ], [ 7, 2 ] ],
  [ [ 6, 1 ], [ 8, 1 ], [ 8, 3 ] ], [ [ 8, 1 ], [ 10, 1 ] ],
  [ [ 9, 1 ], [ 9, 2 ], [ 9, 3 ], [ 10, 1 ] ], [ [ 9, 1 ] ],
  [ [ 9, 1 ] ], [ [ 11, 1 ] ], [ [ 11, 1 ] ], [ [ 11, 1 ] ], [  ] ]
gap> last[3];
[ [ 6, 1 ], [ 8, 1 ], [ 8, 3 ] ]
gap> u5:=ClassElementLattice(ConjugacyClassesSubgroups(l)[8],1);
Group([ (3,4), (2,4,3) ])
gap> u6:=ClassElementLattice(ConjugacyClassesSubgroups(l)[8],3);
Group([ (1,3), (1,3,4) ])
gap> IsSubgroup(u5,u2);
true
gap> IsSubgroup(u6,u2);
true

39.20-6 LowLayerSubgroups

This function computes representatives of the conjugacy classes of subgroups of the finite group G such that the subgroups can be obtained as lim-fold iterated maximal subgroups. If a function cond is given, only subgroups for which this function returns true (also for their intermediate overgroups) is returned. If also a function dosub is given, maximal subgroups are only attempted if this function returns true (this is separated for performance reasons). In the example below, the result would be the same with leaving out the fourth function, but calculation this way is slightly faster. If an initial argument act is given, it must be a group containing and normalizing G, and representatives for classes under the action of this group are chosen.

gap> g:=SymmetricGroup(12);;
gap> l:=LowLayerSubgroups(g,2,x->Size(x)>100000,x->Size(x)>200000);;
gap> Collected(List(l,Size));
[ [ 100800, 1 ], [ 120960, 1 ], [ 161280, 1 ], [ 241920, 1 ], [ 302400, 3 ],
  [ 322560, 1 ], [ 483840, 3 ], [ 518400, 3 ], [ 604800, 1 ], [ 725760, 1 ],
  [ 967680, 1 ], [ 1036800, 1 ], [ 1088640, 3 ], [ 2177280, 1 ],
  [ 3628800, 3 ], [ 7257600, 1 ], [ 19958400, 1 ], [ 39916800, 1 ],
  [ 239500800, 1 ], [ 479001600, 1 ] ]

39.20-7 ContainedConjugates

For $A,B \leq G$ this operation returns representatives of the A conjugacy classes of subgroups that are conjugate to B under G. The function returns a list of pairs of subgroup and conjugating element. If the optional fourth argument onlyone is given as true, then only one pair (or fail if none exists) is returned.

gap> g:=SymmetricGroup(8);;
gap> a:=TransitiveGroup(8,47);;b:=TransitiveGroup(8,9);;
gap> ContainedConjugates(g,a,b);
[ [ Group([ (1,8)(2,3)(4,5)(6,7), (1,3)(2,8)(4,6)(5,7), (1,5)(2,6)(3,7)(4,8),
        (4,5)(6,7) ]), () ],
  [ Group([ (1,8)(2,3)(4,5)(6,7), (1,5)(2,6)(3,7)(4,8), (1,3)(2,8)(4,6)(5,7),
        (2,3)(6,7) ]), (2,4)(3,5) ] ]
gap> ContainedConjugates(g,a,b,true);
[ Group([ (1,8)(2,3)(4,5)(6,7), (1,3)(2,8)(4,6)(5,7), (1,5)(2,6)(3,7)(4,8),
    (4,5)(6,7) ]), () ]

39.20-8 ContainingConjugates

For $A,B \leq G$ this operation returns all G conjugates of A that contain B. The function returns a list of pairs of subgroup and conjugating element.

gap> g:=SymmetricGroup(8);;
gap> a:=TransitiveGroup(8,47);;b:=TransitiveGroup(8,7);;
gap> ContainingConjugates(g,a,b);
[ [ Group([ (1,3,5,7), (3,5), (1,4)(2,7)(3,6)(5,8) ]), (2,3,5,4)(7,8) ] ]

39.20-9 MinimalFaithfulPermutationDegree

For a finite group G, MinimalFaithfulPermutationDegree calculates the least positive integer $n=\mu(G)$ such that G is isomorphic to a subgroup of the symmetric group of degree $n$. This can require calculating the whole subgroup lattice. The operation MinimalFaithfulPermutationRepresentation returns a corresponding isomorphism.

gap> MinimalFaithfulPermutationDegree(SmallGroup(96,3));
12
gap> g:=TransitiveGroup(10,32);;
gap> MinimalFaithfulPermutationDegree(g);
6
gap> map:=MinimalFaithfulPermutationRepresentation(g);;
gap> Size(Image(map));
720

39.20-10 RepresentativesPerfectSubgroups

returns a list of conjugacy representatives of perfect (respectively simple) subgroups of G. This uses the library of perfect groups (see PerfectGroup (50.6-2)), thus it will issue an error if the library is insufficient to determine all perfect subgroups.

gap> m11:=TransitiveGroup(11,6);
M(11)
gap> r:=RepresentativesPerfectSubgroups(m11);;
gap> List(r,Size);
[ 60, 60, 360, 660, 7920, 1 ]
gap> List(r,StructureDescription);
[ "A5", "A5", "A6", "PSL(2,11)", "M11", "1" ]

39.20-11 ConjugacyClassesPerfectSubgroups

returns a list of the conjugacy classes of perfect subgroups of G. (see RepresentativesPerfectSubgroups (39.20-10).)

gap> r := ConjugacyClassesPerfectSubgroups(m11);;
gap> List(r, x -> StructureDescription(Representative(x)));
[ "A5", "A5", "A6", "PSL(2,11)", "M11", "1" ]
gap> SortedList( List(r,Size) );
[ 1, 1, 11, 12, 66, 132 ]

39.20-12 Zuppos

The Zuppos of a group are the cyclic subgroups of prime power order. (The name Zuppo derives from the German abbreviation for zyklische Untergruppen von Primzahlpotenzordnung.) This attribute gives generators of all such subgroups of a group G. That is all elements of G of prime power order up to the equivalence that they generate the same cyclic subgroup.

39.20-13 InfoLattice

is the information class used by the cyclic extension methods for subgroup lattice calculations.

39.21 Specific Methods for Subgroup Lattice Computations

39.21-1 LatticeByCyclicExtension

computes the lattice of G using the cyclic extension algorithm. If the function func is given, the algorithm will discard all subgroups not fulfilling func (and will also not extend them), returning a partial lattice. This can be useful to compute only subgroups with certain properties. Note however that this will not necessarily yield all subgroups that fulfill func, but the subgroups whose subgroups are used for the construction must also fulfill func as well. (In fact the filter func will simply discard subgroups in the cyclic extension algorithm. Therefore the trivial subgroup will always be included.) Also note, that for such a partial lattice maximality/minimality inclusion relations cannot be computed. (If func is a list of length 2, its first entry is such a discarding function, the second a function for discarding zuppos.)

The cyclic extension algorithm requires the perfect subgroups of G. However GAP cannot analyze the function func for its implication but can only apply it. If it is known that func implies solvability, the computation of the perfect subgroups can be avoided by giving a third parameter noperf set to true.

gap> g:=WreathProduct(Group((1,2,3),(1,2)),Group((1,2,3,4)));;
gap> l:=LatticeByCyclicExtension(g,function(G)
> return Size(G) in [1,2,3,6];end);
<subgroup lattice of <permutation group of size 5184 with
9 generators>, 47 classes,
2628 subgroups, restricted under further condition l!.func>

The total number of classes in this example is much bigger, as the following example shows:

gap> LatticeSubgroups(g);
<subgroup lattice of <permutation group of size 5184 with
9 generators>, 566 classes, 27134 subgroups>

39.21-2 InvariantSubgroupsElementaryAbelianGroup

Let G be an elementary abelian group and homs be a set of automorphisms of G. Then this function computes all subspaces of G which are invariant under all automorphisms in homs. When considering G as a module for the algebra generated by homs, these are all submodules. If homs is empty, it computes all subgroups. If the optional parameter dims is given, only submodules of this dimension are computed.

gap> g:=Group((1,2,3),(4,5,6),(7,8,9));
Group([ (1,2,3), (4,5,6), (7,8,9) ])
gap> hom:=GroupHomomorphismByImages(g,g,[(1,2,3),(4,5,6),(7,8,9)],
> [(7,8,9),(1,2,3),(4,5,6)]);
[ (1,2,3), (4,5,6), (7,8,9) ] -> [ (7,8,9), (1,2,3), (4,5,6) ]
gap> u:=InvariantSubgroupsElementaryAbelianGroup(g,[hom]);
[ Group([ (7,8,9), (4,5,6), (1,2,3) ]),
  Group([ (1,3,2)(7,8,9), (1,3,2)(4,5,6) ]),
  Group([ (1,2,3)(4,5,6)(7,8,9) ]), Group(()) ]

39.21-3 SubgroupsSolvableGroup

This function (implementing the algorithm published in [Hul99]) computes subgroups of a solvable group G, using the homomorphism principle. It returns a list of representatives up to G-conjugacy.

The optional argument opt is a record, which may be used to suggest restrictions on the subgroups computed. The following record components of opt are recognized and have the following effects. Note that all of the following restrictions to subgroups with particular properties are only used to speed up the calculation, but the result might still contain subgroups (that had to be computed in any case) that do not satisfy the properties. If this is not desired, the calculation must be followed by an explicit test for the desired properties (which is not done by default, as it would be a general slowdown). The function guarantees that representatives of all subgroups that satisfy the properties are found, i.e. there can be only false positives.

gap> g:=Group((1,2,3),(1,2),(4,5,6),(4,5),(7,8,9),(7,8));
Group([ (1,2,3), (1,2), (4,5,6), (4,5), (7,8,9), (7,8) ])
gap> hom:=GroupHomomorphismByImages(g,g,
> [(1,2,3),(1,2),(4,5,6),(4,5),(7,8,9),(7,8)],
> [(4,5,6),(4,5),(7,8,9),(7,8),(1,2,3),(1,2)]);
[ (1,2,3), (1,2), (4,5,6), (4,5), (7,8,9), (7,8) ] ->
[ (4,5,6), (4,5), (7,8,9), (7,8), (1,2,3), (1,2) ]
gap> l:=SubgroupsSolvableGroup(g,rec(actions:=[hom]));;
gap> SortedList(List(l,Size));
[ 1, 2, 3, 4, 6, 8, 9, 18, 27, 54, 108, 216 ]
gap> Length(ConjugacyClassesSubgroups(g)); # to compare
162

39.21-4 SizeConsiderFunction

This function returns a function consider of five arguments that can be used in SubgroupsSolvableGroup (39.21-3) for the option consider to compute subgroups whose sizes are divisible by size.

gap> l:=SubgroupsSolvableGroup(g,rec(actions:=[hom],
> consider:=SizeConsiderFunction(6)));;
gap> SortedList(List(l,Size));
[ 1, 3, 6, 9, 18, 27, 54, 108, 216 ]

This example shows that in general the consider function does not provide a perfect filter. It is guaranteed that all subgroups fulfilling the condition are returned, but not all subgroups returned necessarily fulfill the condition.

39.21-5 ExactSizeConsiderFunction

This function returns a function consider of five arguments that can be used in SubgroupsSolvableGroup (39.21-3) for the option consider to compute subgroups whose sizes are exactly size.

gap> l:=SubgroupsSolvableGroup(g,rec(actions:=[hom],
> consider:=ExactSizeConsiderFunction(6)));;
gap> SortedList(List(l,Size));
[ 1, 3, 6, 9, 27, 54, 108, 216 ]

Again, the consider function does not provide a perfect filter. It is guaranteed that all subgroups fulfilling the condition are returned, but not all subgroups returned necessarily fulfill the condition.

39.21-6 InfoPcSubgroup

39.22 Special Generating Sets

39.22-1 GeneratorsSmallest

returns a smallest generating set for the group G. This is the lexicographically (using GAPs order of group elements) smallest list $l$ of elements of G such that $G = \langle l \rangle$ and $l_i \not \in \langle l_1, \ldots, l_{{i-1}} \rangle$ (in particular $l_1$ is not the identity element of the group). The comparison of two groups via lexicographic comparison of their sorted element lists yields the same relation as lexicographic comparison of their smallest generating sets.

gap> g:=SymmetricGroup(4);;
gap> GeneratorsSmallest(g);
[ (3,4), (2,3), (1,2) ]

39.22-2 LargestElementGroup

returns the largest element of G with respect to the ordering < of the elements family.

39.22-3 MinimalGeneratingSet

Note that only methods for finite groups, solvable groups, or finitely generated nilpotent groups are available (the latter through the Polycyclic package) and that calculations for nonsolvable finite groups of higher rank can be expensive.

If you do not really need a minimal generating set, but are satisfied with getting a reasonably small set of generators, you better use SmallGeneratingSet (39.22-4).

Information about the minimal generating sets of the finite simple groups of order less than $10^6$ can be found in [MY79]. See also the package AtlasRep.

gap> MinimalGeneratingSet(g);
[ (2,4,3), (1,4,2,3) ]

39.22-4 SmallGeneratingSet

returns a generating set of G which has few elements. As neither irredundancy, nor minimal length is proven it runs much faster than MinimalGeneratingSet (39.22-3). It can be used whenever a short generating set is desired which not necessarily needs to be optimal.

gap> SmallGeneratingSet(g);
[ (1,2,3,4), (1,2) ]

39.22-5 IndependentGeneratorsOfAbelianGroup

returns a list of generators $a_1, a_2, \ldots$ of prime power order or infinite order of the abelian group A such that A is the direct product of the cyclic groups generated by the $a_i$. The list of orders of the returned generators must match the result of AbelianInvariants (39.16-1) (taking into account that zero and infinity (18.2-1) are identified).

gap> g:=AbelianGroup(IsPermGroup,[15,14,22,78]);;
gap> List(IndependentGeneratorsOfAbelianGroup(g),Order);
[ 2, 2, 2, 3, 3, 5, 7, 11, 13 ]
gap> AbelianInvariants(g);
[ 2, 2, 2, 3, 3, 5, 7, 11, 13 ]

39.22-6 IndependentGeneratorExponents

For an abelian group G, with IndependentGeneratorsOfAbelianGroup (39.22-5) value the list $[ a_1, \ldots, a_n ]$, this operation returns the exponent vector $[ e_1, \ldots, e_n ]$ to represent $\textit{g} = \prod_i a_i^{{e_i}}$.

gap> g := AbelianGroup([16,9,625]);;
gap> gens := IndependentGeneratorsOfAbelianGroup(g);;
gap> List(gens, Order);
[ 9, 16, 625 ]
gap> AbelianInvariants(g);
[ 9, 16, 625 ]
gap> r:=gens[1]^4*gens[2]^12*gens[3]^128;;
gap> IndependentGeneratorExponents(g,r);
[ 4, 12, 128 ]

39.23 1-Cohomology

Let $G$ be a finite group and $M$ an elementary abelian normal $p$-subgroup of $G$. Then the group of 1-cocycles $Z^1( G/M, M )$ is defined as

\[ Z^1(G/M, M) = \{ \gamma: G/M \rightarrow M \mid \forall g_1, g_2 \in G : \gamma(g_1 M \cdot g_2 M ) = \gamma(g_1 M)^{{g_2}} \cdot \gamma(g_2 M) \} \]

\[ B^1(G/M, M) = \{ \gamma : G/M \rightarrow M \mid \exists m \in M \forall g \in G : \gamma(gM) = (m^{{-1}})^g \cdot m \} \]

Let $\alpha$ be the isomorphism of $M$ into a row vector space ${\cal W}$ and $(g_1, \ldots, g_l)$ representatives for a generating set of $G/M$. Then there exists a monomorphism $\beta$ of $Z^1( G/M, M )$ in the $l$-fold direct sum of ${\cal W}$, such that $\beta( \gamma ) = ( \alpha( \gamma(g_1 M) ),\ldots, \alpha( \gamma(g_l M) ) )$ for every $\gamma \in Z^1( G/M, M )$.

39.23-1 OneCocycles

Computes the group of 1-cocycles $Z^1(\textit{G}/\textit{M},\textit{M})$. The normal subgroup M may be given by a (Modulo)Pcgs mpcgs. In this case the whole calculation is performed modulo the normal subgroup defined by DenominatorOfModuloPcgs(mpcgs) (see 45.1). Similarly the group G may instead be specified by a set of elements gens that are representatives for a generating system for the factor group G/M. If this is done the 1-cocycles are computed with respect to these generators (otherwise the routines try to select suitable generators themselves). The current version of the code assumes that G is a permutation group or a pc group.

39.23-2 OneCoboundaries

computes the group of 1-coboundaries. Syntax of input and output otherwise is the same as with OneCocycles (39.23-1) except that entries that refer to cocycles are not computed.

The operations OneCocycles (39.23-1) and OneCoboundaries return a record with (at least) the components:

If the factor G/M is given by a (modulo) pcgs gens then special methods are used that compute a presentation for the factor implicitly from the pcgs.

Note that the groups of 1-cocycles and 1-coboundaries are not groups in the sense of Group (39.2-1) for GAP but vector spaces.

gap> g:=Group((1,2,3,4),(1,2));;
gap> n:=Group((1,2)(3,4),(1,3)(2,4));;
gap> oc:=OneCocycles(g,n);
rec( cocycleToComplement := function( c ) ... end,
  cocycleToList := function( c ) ... end,
  complement := Group([ (3,4), (2,4,3) ]),
  complementGens := [ (3,4), (2,4,3) ],
  complementToCocycle := function( K ) ... end,
  factorGens := [ (3,4), (2,4,3) ], generators := [ (3,4), (2,4,3) ],
  isSplitExtension := true, listToCocycle := function( L ) ... end,
  oneCoboundaries := <vector space over GF(2), with 2 generators>,
  oneCocycles := <vector space over GF(2), with 2 generators> )
gap> oc.cocycleToList([ 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0 ]);
[ (1,2)(3,4), (1,2)(3,4) ]
gap> oc.listToCocycle([(),(1,3)(2,4)]) = Z(2) * [ 0, 0, 1, 0];
true
gap> oc.cocycleToComplement([ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ]);
Group([ (3,4), (1,3,4) ])
gap> oc.complementToCocycle(Group((1,2,4),(1,4))) = Z(2) * [ 0, 1, 1, 1 ];
true

The factor group $H^1(\textit{G}/\textit{M}, \textit{M}) = Z^1(\textit{G}/\textit{M}, \textit{M}) / B^1(\textit{G}/\textit{M}, \textit{M})$ is called the first cohomology group. Currently there is no function which explicitly computes this group. The easiest way to represent it is as a vector space complement to $B^1$ in $Z^1$.

If the only purpose of the calculation of $H^1$ is the determination of complements it might be desirable to stop calculations once it is known that the extension cannot split. This can be achieved via the more technical function OCOneCocycles (39.23-3).

39.23-3 OCOneCocycles

is the more technical function to compute 1-cocycles. It takes a record ocr as first argument which must contain at least the components group for the group and modulePcgs for a (modulo) pcgs of the module. This record will also be returned with components as described under OneCocycles (39.23-1) (with the exception of isSplitExtension which is indicated by the existence of a complement) but components such as oneCoboundaries will only be computed if not already present.

If onlySplit is true, OCOneCocycles returns false as soon as possible if the extension does not split.

39.23-4 ComplementClassesRepresentativesEA

computes complement classes to an elementary abelian normal subgroup N via 1-Cohomology. Normally, a user program should call ComplementClassesRepresentatives (39.11-6) instead, which also works for a solvable (not necessarily elementary abelian) N.

39.23-5 InfoCoh

39.24 Schur Covers and Multipliers

Additional attributes and properties of a group can be derived from computing its Schur cover. For example, if $G$ is a finitely presented group, the derived subgroup of a Schur cover of $G$ is invariant and isomorphic to the NonabelianExteriorSquare (39.24-5) value of $G$, see [BJR87].

39.24-1 EpimorphismSchurCover

returns an epimorphism $epi$ from a group $D$ onto G. The group $D$ is one (of possibly several) Schur covers of G. The group $D$ can be obtained as the Source (32.3-8) value of epi. The kernel of $epi$ is the Schur multiplier of G. If pl is given as a list of primes, only the multiplier part for these primes is realized. At the moment, $D$ is represented as a finitely presented group.

39.24-2 SchurCover

At the moment this cover is represented as a finitely presented group and IsomorphismPermGroup (43.3-1) would be needed to convert it to a permutation group.

If also the relation to G is needed, EpimorphismSchurCover (39.24-1) should be used.

gap> g:=Group((1,2,3,4),(1,2));;
gap> epi:=EpimorphismSchurCover(g);
[ F1, F2, F3 ] -> [ (1,2), (2,3), (3,4) ]
gap> Size(Source(epi));
48
gap> f:=FreeGroup("a","b");;
gap> g:=f/ParseRelators(f,"a2,b3,(ab)5");;
gap> epi:=EpimorphismSchurCover(g);
[ a, b ] -> [ a, b ]
gap> Size(Kernel(epi));
2

There is another operation, AbelianInvariantsMultiplier (39.24-3), which only returns the structure of the Schur Multiplier, and which should work for larger groups as well.

39.24-3 AbelianInvariantsMultiplier

At the moment, this operation will not give any information about how to extend the multiplier to a Schur Cover.

gap> AbelianInvariantsMultiplier(g);
[ 2 ]
gap> AbelianInvariantsMultiplier(AlternatingGroup(6));
[ 2, 3 ]
gap> AbelianInvariantsMultiplier(SL(2,3));
[  ]
gap> AbelianInvariantsMultiplier(SL(3,2));
[ 2 ]
gap> AbelianInvariantsMultiplier(PSU(4,2));
[ 2 ]

The GAP 4.4.12 manual contained examples for larger groups e.g. $M_{22}$. However, some issues that may very rarely (and not easily reproducibly) lead to wrong results were discovered in the code capable of handling larger groups, and in GAP 4.5 it was replaced by a more reliable basic method. To deal with larger groups, one can use the function SchurMultiplier (cohomolo: SchurMultiplier) from the cohomolo package. Also, additional methods for AbelianInvariantsMultiplier are installed in the Polycyclic package for pcp-groups.

39.24-4 Epicentre

There are various ways of describing the epicentre of a group G. It is the smallest normal subgroup $N$ of G such that $\textit{G}/N$ is a central quotient of a group. It is also equal to the Exterior Center of G, see [Ell98].

39.24-5 NonabelianExteriorSquare

Computes the nonabelian exterior square $\textit{G} \wedge \textit{G}$ of the group G, which for a finitely presented group is the derived subgroup of any Schur cover of G (see [BJR87]).

39.24-6 EpimorphismNonabelianExteriorSquare

Computes the mapping $\textit{G} \wedge \textit{G} \rightarrow \textit{G}$. The kernel of this mapping is equal to the Schur multiplier of G.

39.24-7 IsCentralFactor

This function determines if there exists a group $H$ such that G is isomorphic to the quotient $H/Z(H)$. A group with this property is called in literature capable. A group being capable is equivalent to the epicentre of G being trivial, see [BFS79].

39.24-8 Covering groups of symmetric groups

The covering groups of symmetric groups were classified in [Sch11]; an inductive procedure to construct faithful, irreducible representations of minimal degree over all fields was presented in [Maa10]. Methods for EpimorphismSchurCover (39.24-1) are provided for natural symmetric groups which use these representations. For alternating groups, the restriction of these representations are provided, but they may not be irreducible. In the case of degree $6$ and $7$, they are not the full covering groups and so matrix representations are just stored explicitly for the six-fold covers.

gap> EpimorphismSchurCover(SymmetricGroup(15));
[ < immutable compressed matrix 64x64 over GF(9) >,
  < immutable compressed matrix 64x64 over GF(9) > ] ->
[ (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15), (1,2) ]
gap> EpimorphismSchurCover(AlternatingGroup(15));
[ < immutable compressed matrix 64x64 over GF(9) >,
  < immutable compressed matrix 64x64 over GF(9) > ] ->
[ (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15), (13,14,15) ]
gap> SchurCoverOfSymmetricGroup(12);
<matrix group of size 958003200 with 2 generators>
gap> DoubleCoverOfAlternatingGroup(12);
<matrix group of size 479001600 with 2 generators>
gap> BasicSpinRepresentationOfSymmetricGroup( 10, 3, -1 );
[ < immutable compressed matrix 16x16 over GF(9) >,
  < immutable compressed matrix 16x16 over GF(9) >,
  < immutable compressed matrix 16x16 over GF(9) >,
  < immutable compressed matrix 16x16 over GF(9) >,
  < immutable compressed matrix 16x16 over GF(9) >,
  < immutable compressed matrix 16x16 over GF(9) >,
  < immutable compressed matrix 16x16 over GF(9) >,
  < immutable compressed matrix 16x16 over GF(9) >,
  < immutable compressed matrix 16x16 over GF(9) > ]

39.24-9 BasicSpinRepresentationOfSymmetricGroup

Constructs the image of the Coxeter generators in the basic spin (projective) representation of the symmetric group of degree n over a field of characteristic $\textit{p} \geq 0$. There are two such representations and sign controls which is returned: +1 gives a group where the preimage of an adjacent transposition $(i,i+1)$ has order 4, -1 gives a group where the preimage of an adjacent transposition $(i,i+1)$ has order 2. If no sign is specified, +1 is used by default. If no p is specified, 3 is used by default. (Note that the convention of which cover is labelled as +1 is inconsistent in the literature.)

39.24-10 SchurCoverOfSymmetricGroup

Constructs a Schur cover of SymmetricGroup(n) as a faithful, irreducible matrix group in characteristic p ($\textit{p} \neq 2$). For $\textit{n} \geq 4$, there are two such covers, and sign determines which is returned: +1 gives a group where the preimage of an adjacent transposition $(i,i+1)$ has order 4, -1 gives a group where the preimage of an adjacent transposition $(i,i+1)$ has order 2. If no sign is specified, +1 is used by default. If no p is specified, 3 is used by default. (Note that the convention of which cover is labelled as +1 is inconsistent in the literature.) For $\textit{n} \leq 3$, the symmetric group is its own Schur cover and sign is ignored. For $\textit{p} = 2$, there is no faithful, irreducible representation of the Schur cover unless $\textit{n} = 1$ or $\textit{n} = 3$, so fail is returned if $\textit{p} = 2$. For $\textit{p} = 3$, $\textit{n} = 3$, the representation is indecomposable, but reducible. The field of the matrix group is generally GF(p^2) if $\textit{p} > 0$, and an abelian number field if $\textit{p} = 0$.

39.24-11 DoubleCoverOfAlternatingGroup

Constructs a double cover of AlternatingGroup(n) as a faithful, completely reducible matrix group in characteristic p ($p \neq 2$) for $n \geq 4$. For $n \leq 3$, the alternating group is its own Schur cover, and fail is returned. For $p = 2$, there is no faithful, completely reducible representation of the double cover, so fail is returned. The field of the matrix group is generally GF(p^2) if $p>0$, and an abelian number field if $p=0$. If p is omitted, the default is 3.

39.25 2-Cohomology

39.25-1 TwoCohomologyGeneric

This function computes the second cohomology group for an arbitrary finite group G. The generators of the module M must correspond to the generators of G. It returns a record with components coboundaries, cocycles and cohomology which are lists of vectors that form a basis of the respective group. cohomology is chosen as a vector space complement to coboundaries in the cocycles. These vectors are representing tails in M with respect to the relators of the presentation presentation of G. (Note that this presentation is on a generating set chosen by the routine, this generating system corresponds to the components group and module of the record returned. The extension corresponding to a cocyle c can be constructed as FpGroupCocycle(r,c) where r is the cohomology record. This is currently done as a finitely presented group.

gap> g:=Group((1,2,3,4,5),(1,2,3));;
gap> mats:=[[[2,0,0,1],[1,2,1,0],[2,1,1,1],[2,1,1,0]],
> [[0,2,0,0],[1,2,1,0],[0,0,1,0],[0,0,0,1]]]*Z(3)^0;;
gap> mo:=GModuleByMats(mats,GF(3));;
gap> coh:=TwoCohomologyGeneric(g,mo);;
gap> coh.cocycles;
< immutable compressed matrix 8x44 over GF(3) >
gap> coh.coboundaries;
[ < immutable compressed vector length 44 over GF(3) >,
  < immutable compressed vector length 44 over GF(3) >,
  < immutable compressed vector length 44 over GF(3) >,
  < immutable compressed vector length 44 over GF(3) >,
  < immutable compressed vector length 44 over GF(3) >,
  < immutable compressed vector length 44 over GF(3) >,
  < immutable compressed vector length 44 over GF(3) > ]
gap> coh.cohomology;
[ < immutable compressed vector length 44 over GF(3) > ]
gap> g1:=FpGroupCocycle(coh,coh.zero,true);
<fp group of size 4860 on the generators [ F1, F2, F3, m1, m2, m3, m4 ]>
gap> g2:=FpGroupCocycle(coh,coh.cohomology[1],true);
<fp group of size 4860 on the generators [ F1, F2, F3, m1, m2, m3, m4 ]>
gap> g1:=Image(IsomorphismPermGroup(g1));
<permutation group with 7 generators>
gap> Length(ComplementClassesRepresentatives(g1,SolvableRadical(g1)));
3
gap> g2:=Image(IsomorphismPermGroup(g2));
<permutation group with 7 generators>
gap> Length(ComplementClassesRepresentatives(g2,SolvableRadical(g2)));
0

39.25-2 FpGroupCocycle

For a record r as returned by TwoCohomologyGeneric (39.25-1) and a vector c in the space of two-cocycles, this operation returns a finitely presented group that is an extension corresponding to the cocycle c. If the underlying module has dimension $d$, the last $d$ generators generate the normal subgroup. If the optional parameter doperm is given as true, a faithful permutation representation is computed and stored in the attribute IsomorphismPermGroup (43.3-1) of the computed group. If the option normalform is given as true, arithmetic in the resulting finitely presented group will bring words into normal form.

gap> g:=Group((2,15,8,16)(3,17,14,21)(4,23,20,6)(5,9,22,11)(7,13,19,25),
> (2,12,7,17)(3,18,13,23)(4,24,19,9)(5,10,25,15)(6,11,16,21));;
gap> StructureDescription(g);
"GL(2,5)"
gap> mats:=[[[1,1,0,2],[2,0,0,0],[0,2,2,0],[0,1,0,0]],
> [[0,0,0,1],[1,1,2,0],[1,0,2,1],[1,0,1,0]]]*Z(3)^0;;
gap> mo:=GModuleByMats(mats,GF(3));;
gap> coh:=TwoCohomologyGeneric(g,mo);;
gap> coh.cohomology;
[ < immutable compressed vector length 116 over GF(3) > ]
gap> p:=FpGroupCocycle(coh,coh.zero,true);
<fp group of size 38880 on the generators
[ F1, F2, F3, F4, F5, F6, m1, m2, m3, m4 ]>
gap> g1:=Image(IsomorphismPermGroup(p));
<permutation group with 10 generators>
gap> p:=FpGroupCocycle(coh,coh.cohomology[1],true);
<fp group of size 38880 on the generators
[ F1, F2, F3, F4, F5, F6, m1, m2, m3, m4 ]>
gap> g2:=Image(IsomorphismPermGroup(p));
<permutation group with 10 generators>
gap> Collected(List(MaximalSubgroupClassReps(g1),Size));
[ [ 480, 3 ], [ 3888, 1 ], [ 6480, 1 ], [ 7776, 1 ], [ 19440, 1 ] ]
gap> Collected(List(MaximalSubgroupClassReps(g2),Size));
[ [ 3888, 1 ], [ 6480, 1 ], [ 7776, 1 ], [ 19440, 1 ] ]
gap> p:=FpGroupCocycle(coh,coh.cohomology[1],true:normalform);;

gap> p.7*p.1; # i.e. m1*F1, but in normal form
F1*m4

39.26 Tests for the Availability of Methods

The following filters and operations indicate capabilities of GAP. They can be used in the method selection or algorithms to check whether it is feasible to compute certain operations for a given group. In general, they return true if good algorithms for the given arguments are available in GAP. An answer false indicates that no method for this group may exist, or that the existing methods might run into problems.

Typical examples when this might happen is with finitely presented groups, for which many of the methods cannot be guaranteed to succeed in all situations.

The willingness of GAP to perform certain operations may change, depending on which further information is known about the arguments. Therefore the filters used are not implemented as properties but as other filters (see 13.7 and 13.8).

39.26-1 CanEasilyTestMembership

This filter indicates whether GAP can test membership of elements in the group G (via the operation \in (30.6-1)) in reasonable time. It is used by the method selection to decide whether an algorithm that relies on membership tests may be used.

39.26-2 CanEasilyComputeWithIndependentGensAbelianGroup

This filter indicates whether GAP can in reasonable time compute independent abelian generators of the group G (via IndependentGeneratorsOfAbelianGroup (39.22-5)) and then can decompose arbitrary group elements with respect to these generators using IndependentGeneratorExponents (39.22-6). It is used by the method selection to decide whether an algorithm that relies on these two operations may be used.

39.26-3 CanComputeSize

This filter indicates that we know that the size of the domain dom (which might be infinity (18.2-1)) can be computed reasonably easily. It doesn't imply as quick a computation as HasSize would but its absence does not imply that the size cannot be computed.

39.26-4 CanComputeSizeAnySubgroup

This filter indicates whether GAP can easily compute the size of any subgroup of the group G. (This is for example advantageous if one can test that a stabilizer index equals the length of the orbit computed so far to stop early.)

39.26-5 CanComputeIndex

This function indicates whether the index $[\textit{G}:\textit{H}]$ (which might be infinity (18.2-1)) can be computed. It assumes that $\textit{H} \leq \textit{G}$ (see CanComputeIsSubset (39.26-6)).

39.26-6 CanComputeIsSubset

This filter indicates that GAP can test (via IsSubset (30.5-1)) whether B is a subset of A.

39.26-7 KnowsHowToDecompose

Tests whether the group G can decompose elements in the generators gens. If gens is not given it tests, whether it can decompose in the generators given in the GeneratorsOfGroup (39.2-4) value of G.

This property can be used for example to check whether a group homomorphism by images (see GroupHomomorphismByImages (40.1-1)) can be reasonably defined from this group.

39.27 Specific functions for Normalizer calculation

39.27-1 NormalizerViaRadical

This function implements a particular approach, following the SolvableRadical paradigm, for calculating the normalizer of a subgroup S in G. It is at the moment provided only as a separate function, and not as method for the operation Normalizer, as it can often be slower than other built-in routines. In certain hard cases (non-solvable groups with nontrivial radical), however its performance is substantially superior. The function thus is provided as a non-automated tool for advanced users.

gap> g:=TransitiveGroup(30,2030);;
gap> s:=SylowSubgroup(g,5);;
gap> Size(NormalizerViaRadical(g,s));
28800

Note that this example only demonstrates usage, but that in this case in fact the ordinary Normalizer routine performs faster.

39 Groups