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1 The GAP Table of Marks Library

1.1 Tables Of Marks

The concept of a Table of Marks was introduced by W.Burnside in his book ``Theory of Groups of Finite Order'' [Bur55]. Therefore a table of marks is sometimes called a Burnside matrix. The table of marks of a finite group G is a matrix whose rows and columns are labelled by the conjugacy classes of subgroups of G and where for two subgroups H and K the (H, K)–entry is the number of fixed points of K in the transitive action of G on the cosets of H in G. So the table of marks characterizes the set of all permutation representations of G. Moreover, the table of marks gives a compact description of the subgroup lattice of G, since from the numbers of fixed points the numbers of conjugates of a subgroup K contained in a subgroup H can be derived. For small groups the table of marks of G can be constructed directly in GAP by first computing the entire subgroup lattice of G. However, for larger groups this method is unfeasible. The GAP Table of Marks library provides access to several hundred table of marks and their maximal subgroups.

1.2 Installing The Table of Marks Library

Download the archives in your preferred format. Unpack the archives inside the pkg dirctory of your GAP installation. Load the package

gap> LoadPackage("tomlib");
true

1.3 Contents

TomLib contains several hundred tables of marks. For a complete list of the contents of the library do the following.

gap> names:=AllLibTomNames();;
gap> "A5" in names;
true

The current version of the tomlib contains the tables of marks of the groups listed below as well as the tables of many of their maximal subgroups and automorphism groups. The Alternating groups A_n

• for n = 5, 6, 7, 8, 9, 10, 11, 12, 13.

The Symmetric groups S_n

• for n = 4, 5, 6, 7, 8, 9, 10, 11, 12, 13.

The Linear groups L_2(n) for

• n = 7, 8, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53

• n = 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125 .

along with

• L_3(4), L_3(3), L_3(5), L_3(7), L_3(9)

• L_4(3), L_3(8), L_3(11).

The Unitary groups

• U_3(3), U_4(3), U_3(5), U_3(4), U_3(11), U_3(7), U_3(8)

• U_3(9), U_4(2), U_5(2)

The Sporadic Groups

• Co_3, HS, McL, He, J_1, J_2, J_3, M_11, M_12, M_22, M_23, M_24

The names given to each subgroup are consistent with those used in Robert Wilson's atlas [WWT+] For example if you wish to access the table of marks of the maximal subgroup "5:4 × A5" of the Higman-Sims group do the following:

gap> TableOfMarks("5:4xA5");
TableOfMarks( "5:4xA5" )

1.4 Administrative Functions

Here we document some of the administrative facilities for the the GAP library of tables of marks.

1.4-1 LIBTOMKNOWN

LIBTOMKNOWN is a record that controls the loading of data files of the GAP library of tables of marks. It has the following components.

ACTUAL

the name of the file to be read at the moment (set by SetActualLibFileName),

LOADSTATUS

a record whose components are names of files in the library of tables of marks, with value a list whose first entry is one of "loaded", "unloaded", "userloaded" and whose second entry is an integer that controls when the corresponding tables of marks can be removed from GAP,

MAX

GAP remembers at most MAX of the previously loaded library files (the default value is 10),

UNLOAD

is it allowed to remove previously loaded library files (is set to true by default),

Additionally the names of the files already loaded occur as components of LIBTOMKNOWN; the corresponding values are given by the data of the files.

1.4-2 IsLibTomRep

Library tables of marks have their own representation. IsLibTomRep tests if obj is a library representation.

1.4-3 TableOfMarksFromLibrary

Returns: the table of marks with name string.

1.4-4 ConvertToLibTom

ConvertToLibTom converts a record with components from TableOfMarksComponents into a library table of marks object with the corresponding attribute values set.

1.4-5 SetActualLibFileName

Sets the file name for filename.

1.4-6 LIBTOM

The library format of a library table of marks is a call to the function LIBTOM, with the arguments sorted as in TableOfMarksComponents .

AFLT adds a fusion map value from the table of marks with name source to the table of marks with name destination. The fusion map is a list of positive integers, storing at position i the position of the class in destination that contains the subgroups in the i-th class of source.

ACLT adds the value value of a component with name component to the table of marks with identifier value identifier in the library of tables of marks.

1.4-7 AllLibTomNames

Returns: a list containing all names of available library tables of marks.

1.4-8 NamesLibTom

Returns: all names of the library table tom or of the library table with name string

1.4-9 NotifiedFusionsOfLibTom

Are there any fusions from the library table of marks tom or the table of marks with name string into other library tables marks?

NotifiedFusionsOfLibTom returns the names of all such tables of marks. FusionsOfLibTom returns the complete fusion maps. For that the corresponding library file has to be loaded.

1.4-10 NotifiedFusionsToLibTom

Are there any fusions from other library table of marks to tom or the table of marks with name string.

NotifiedFusionsToLibTom returns the names of all such tables of marks. FusionsToLibTom returns the complete fusion maps. For that, the correponding library files have to be loaded.

1.4-11 UnloadTableOfMarksData

UnloadTableOfMarksData clears the cache of tables of marks. This can be used to free up to several hundred megabytes of space in the current GAP session.

1.5 Standard Generators of Groups

An s-tuple of standard generators of a given group G is a vector (g_1, g_2, ..., g_s) of elements g_i ∈ G satisfying certain conditions (depending on the isomorphism type of G) such that

1. ⟨ g_1, g_2, ..., g_s ⟩ = G and

2. the vector is unique up to automorphisms of G, i.e., for two vectors (g_1, g_2, ..., g_s) and (h_1, h_2, ..., h_s) of standard generators, the map g_i ↦ h_i extends to an automorphism of G.

For details about standard generators, see [Wil96].

1.5-1 StandardGeneratorsInfo

When called with the group G, StandardGeneratorsInfo returns a list of records with at least one of the components script and description. Each such record defines standard generators of groups isomorphic to G, the i-th record is referred to as the i-th set of standard generators for such groups. The value of script is a dense list of lists, each encoding a command that has one of the following forms.

A definition [ i, n, k ] or [ i, n ]

means to search for an element of order n, and to take its k-th power as candidate for the i-th standard generator (the default for k is 1),

a relation [ i_1, k_1, i_2, k_2, ..., i_m, k_m, n ] with m > 1

means a check whether the element g_{i_1}^{k_1} g_{i_2}^{k_2} ⋯ g_{i_m}^{k_m} has order n; if g_j occurs then of course the j-th generator must have been defined before,

a relation [ [ i_1, i_2, ..., i_m ], slp, n ]

means a check whether the result of the straight line program slp (see Reference: Straight Line Programs) applied to the candidates g_{i_1}, g_{i_2}, ..., g_{i_m} has order n, where the candidates g_j for the j-th standard generators must have been defined before,

a condition [ [ i_1, k_1, i_2, k_2, ..., i_m, k_m ], f, v ]

means a check whether the GAP function in the global list StandardGeneratorsFunctions (1.5-3) that is followed by the list f of strings returns the value v when it is called with G and g_{i_1}^{k_1} g_{i_2}^{k_2} ⋯ g_{i_m}^{k_m}.

Optional components of the returned records are

generators

a string of names of the standard generators,

description

a string describing the script information in human readable form, in terms of the generators value,

classnames

a list of strings, the i-th entry being the name of the conjugacy class containing the i-th standard generator, according to the Atlas character table of the group (see ClassNames (Reference: ClassNames)), and

ATLAS

a boolean; true means that the standard generators coincide with those defined in Rob Wilson's Atlas of Group Representations (see [WWT+]), and false means that this property is not guaranteed.

standardization

a positive integer i; Whenever ATLAS returns true the value of i means that the generators stored in the group are standard generators w.r.t. standardization i, in the sense of Rob Wilson's Atlas of Group Representations.

There is no default method for an arbitrary isomorphism type, since in general the definition of standard generators is not obvious.

The function StandardGeneratorsOfGroup (1.5-5) can be used to find standard generators of a given group isomorphic to G.

The generators and description values, if not known, can be computed by HumanReadableDefinition (1.5-2).

gap> StandardGeneratorsInfo( TableOfMarks( "L3(3)" ) );
[ rec( ATLAS := true, 
    description := "|a|=2, |b|=3, |C(b)|=9, |ab|=13, |ababb|=4", 
    generators := "a, b", 
    script := [ [ 1, 2 ], [ 2, 3 ], [ [ 2, 1 ], [ "|C(",, ")|" ], 9 ], 
        [ 1, 1, 2, 1, 13 ], [ 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 4 ] ], 
    standardization := 1 ) ]

1.5-2 HumanReadableDefinition

Let info be a record that is valid as value of StandardGeneratorsInfo (1.5-1). HumanReadableDefinition returns a string that describes the definition of standard generators given by the script component of info in human readable form. The names of the generators are taken from the generators component (default names "a", "b" etc. are computed if necessary), and the result is stored in the description component.

ScriptFromString does the converse of HumanReadableDefinition, i.e., it takes a string string as returned by HumanReadableDefinition, and returns a corresponding script list.

If "condition" lines occur in the script (see StandardGeneratorsInfo (1.5-1)) then the functions that occur must be contained in StandardGeneratorsFunctions (1.5-3).

gap> scr:= ScriptFromString( "|a|=2, |b|=3, |C(b)|=9, |ab|=13, |ababb|=4" );
[ [ 1, 2 ], [ 2, 3 ], [ [ 2, 1 ], [ "|C(",, ")|" ], 9 ], [ 1, 1, 2, 1, 13 ], 
  [ 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 4 ] ]
gap> info:= rec( script:= scr );
rec( script := [ [ 1, 2 ], [ 2, 3 ], [ [ 2, 1 ], [ "|C(",, ")|" ], 9 ], 
      [ 1, 1, 2, 1, 13 ], [ 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 4 ] ] )
gap> HumanReadableDefinition( info );
"|a|=2, |b|=3, |C(b)|=9, |ab|=13, |ababb|=4"
gap> info;
rec( description := "|a|=2, |b|=3, |C(b)|=9, |ab|=13, |ababb|=4", 
  generators := "a, b", 
  script := [ [ 1, 2 ], [ 2, 3 ], [ [ 2, 1 ], [ "|C(",, ")|" ], 9 ], 
      [ 1, 1, 2, 1, 13 ], [ 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 4 ] ] )

1.5-3 StandardGeneratorsFunctions

StandardGeneratorsFunctions is a list of even length. At position 2i-1, a function of two arguments is stored, which are expected to be a group and a group element. At position 2i a list of strings is stored such that first inserting a generator name in all holes and then forming the concatenation yields a string that describes the function at the previous position; this string must contain the generator enclosed in round brackets ( and ).

This list is used by the functions StandardGeneratorsInfo (1.5-1)), HumanReadableDefinition (1.5-2), and ScriptFromString (1.5-2). Note that the lists at even positions must be pairwise different.

gap> StandardGeneratorsFunctions{ [ 1, 2 ] };
[ function( G, g ) ... end, [ "|C(",, ")|" ] ]

1.5-4 IsStandardGeneratorsOfGroup

Let info be a record that is valid as value of StandardGeneratorsInfo (1.5-1), G a group, and gens a list of generators for G. In this case, IsStandardGeneratorsOfGroup returns true if gens satisfies the conditions of the script component of info, and false otherwise.

Note that the result true means that gens is a list of standard generators for G only if G has the isomorphism type for which info describes standard generators.

1.5-5 StandardGeneratorsOfGroup

Let info be a record that is valid as value of StandardGeneratorsInfo (1.5-1), and G a group of the isomorphism type for which info describes standard generators. In this case, StandardGeneratorsOfGroup returns a list of standard generators of G, see Section 1.5.

The optional argument randfunc must be a function that returns an element of G when called with G; the default is PseudoRandom (Reference: PseudoRandom).

In each call to StandardGeneratorsOfGroup, the script component of info is scanned line by line. randfunc is used to find an element of the prescribed order whenever a definition line is met, and for the relation and condition lines in the script list, the current generator candidates are checked; if a condition is not fulfilled, all candidates are thrown away, and the procedure starts again with the first line. When the conditions are fulfilled after processing the last line of the script list, the standard generators are returned.

Note that if G has the wrong isomorphism type then StandardGeneratorsOfGroup returns a list of elements in G that satisfy the conditions of the script component of info if such elements exist, and does not terminate otherwise. In the former case, obviously the returned elements need not be standard generators of G.

gap> a5:= AlternatingGroup( 5 );
Alt( [ 1 .. 5 ] )
gap> info:= StandardGeneratorsInfo( TableOfMarks( "A5" ) )[1];
rec( ATLAS := true, description := "|a|=2, |b|=3, |ab|=5", 
generators := "a, b", script := [ [ 1, 2 ], [ 2, 3 ], [ 1, 1, 2, 1, 5 ] ], 
standardization := 1 )
gap> IsStandardGeneratorsOfGroup( info, a5, [ (1,3)(2,4), (3,4,5) ] );
true
gap> IsStandardGeneratorsOfGroup( info, a5, [ (1,3)(2,4), (1,2,3) ] );
false
gap> s5:= SymmetricGroup( 5 );;
gap> RepresentativeAction( s5, [ (1,3)(2,4), (3,4,5) ], 
>        StandardGeneratorsOfGroup( info, a5 ), OnPairs ) <> fail;
true

1.5-6 StandardGeneratorsInfo

For a table of marks tom, a stored value of StandardGeneratorsInfo equals the value of this attribute for the underlying group (see UnderlyingGroup (Reference: UnderlyingGroup for tables of marks)) of tom, cf. Section 1.5.

In this case, the GeneratorsOfGroup (Reference: GeneratorsOfGroup) value of the underlying group G of tom is assumed to be in fact a list of standard generators for G; So one should be careful when setting the StandardGeneratorsInfo value by hand.

There is no default method to compute the StandardGeneratorsInfo value of a table of marks if it is not yet stored.

gap> a5:=TableOfMarks("A5");
gap> std:= StandardGeneratorsInfo( a5 );
[ rec( ATLAS := true, description := "|a|=2, |b|=3, |ab|=5", 
    generators := "a, b", script := [ [ 1, 2 ], [ 2, 3 ], [ 1, 1, 2, 1, 5 ] 
       ], standardization := 1 ) ]
gap> # Now find standard generators of an isomorphic group.
gap> g:= SL(2,4);;
gap> repeat
>   x:= PseudoRandom( g );
> until Order( x ) = 2;
gap> repeat  
>   y:= PseudoRandom( g );
> until Order( y ) = 3 and Order( x*y ) = 5;
gap> # Compute a representative w.r.t. these generators.
gap> RepresentativeTomByGenerators( a5, 4, [ x, y ] );
Group([ [ [ 0*Z(2), Z(2)^0 ], [ Z(2)^0, 0*Z(2) ] ],
  [ [ Z(2^2), Z(2^2)^2 ], [ Z(2^2)^2, Z(2^2) ] ] ])
gap> # Show that the new generators are really good.
gap> grp:= UnderlyingGroup( a5 );;
gap> iso:= GroupGeneralMappingByImages( grp, g,
>              GeneratorsOfGroup( grp ), [ x, y ] );;
gap> IsGroupHomomorphism( iso );
true
gap> IsBijective( iso );
true

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