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39 Groups
 39.1 Group Elements
 39.2 Creating Groups
 39.3 Subgroups
 39.4 Closures of (Sub)groups
 39.5 Expressing Group Elements as Words in Generators
 39.6 Structure Descriptions
 39.7 Cosets
 39.8 Transversals
 39.9 Double Cosets
 39.10 Conjugacy Classes
 39.11 Normal Structure
 39.12 Specific and Parametrized Subgroups
 39.13 Sylow Subgroups and Hall Subgroups
 39.14 Subgroups characterized by prime powers
 39.15 Group Properties
 39.16 Numerical Group Attributes
 39.17 Subgroup Series
 39.18 Factor Groups
 39.19 Sets of Subgroups
 39.20 Subgroup Lattice
 39.21 Specific Methods for Subgroup Lattice Computations
 39.22 Special Generating Sets
 39.23 1-Cohomology
 39.24 Schur Covers and Multipliers
 39.25 Tests for the Availability of Methods

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, ..., 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
‣ Group( gen, ... )( function )
‣ Group( gens[, id] )( function )

Group( gen, ... ) is the group generated by the arguments gen, ...

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( gens )( operation )
‣ GroupByGenerators( gens, id )( operation )

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( gens[, id] )( operation )

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
‣ GeneratorsOfGroup( G )( attribute )

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
‣ AsGroup( D )( attribute )

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
‣ ConjugateGroup( G, obj )( operation )

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
‣ IsGroup( obj )( category )

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
‣ InfoGroup( info class )

is the info class for the generic group theoretic functions (see 7.4).

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
‣ Subgroup( G, gens )( function )
‣ SubgroupNC( G, gens )( function )
‣ Subgroup( G )( function )

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)
‣ Index( G, U )( operation )
‣ IndexNC( G, U )( operation )

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

gap> Index(g,u);
4

39.3-3 IndexInWholeGroup
‣ IndexInWholeGroup( G )( attribute )

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
‣ AsSubgroup( G, U )( operation )

creates a subgroup of G which contains the same elements as U

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( G, U )( function )

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
‣ IsNormal( G, U )( operation )

returns true if the group G normalizes the group U and false otherwise.

A group G normalizes a group U if and only if for every g ∈ G and u ∈ 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
‣ IsCharacteristicSubgroup( G, N )( operation )

tests whether N is invariant under all automorphisms of G.

gap> IsCharacteristicSubgroup(g,u);
false

39.3-8 ConjugateSubgroup
‣ ConjugateSubgroup( G, g )( operation )

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
‣ ConjugateSubgroups( G, U )( operation )

returns a list of all images of the group U under conjugation action by G.

39.3-10 IsSubnormal
‣ IsSubnormal( G, U )( operation )

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
‣ SubgroupByProperty( G, prop )( function )

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
‣ SubgroupShell( G )( function )

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
‣ ClosureGroup( G, obj )( operation )

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
‣ ClosureGroupAddElm( G, elm )( function )
‣ ClosureGroupCompare( G, elm )( function )
‣ ClosureGroupIntest( G, elm )( function )

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
‣ ClosureGroupDefault( G, elm )( function )

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
‣ ClosureSubgroup( G, obj )( function )
‣ ClosureSubgroupNC( G, obj )( function )

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) is 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
‣ EpimorphismFromFreeGroup( G )( attribute )

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 groups generators.

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
‣ Factorization( G, elm )( operation )

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 group G to the free group in which the word is expressed. 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
‣ GrowthFunctionOfGroup( G )( operation )
‣ GrowthFunctionOfGroup( G, radius )( operation )

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
‣ StructureDescription( G )( attribute )

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>                           ; 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.

The strategy used to compute a StructureDescription is as follows:

1.

Lookup in a precomputed list, if the order of G is not larger than 100 and not equal to 64.

2.

If G is abelian, then decompose it into cyclic factors in "elementary divisors style". For example, "C2 x C3 x C3" is "C6 x C3".

3.

Recognize alternating groups, symmetric groups, dihedral groups, quasidihedral groups, quaternion groups, PSL's, SL's, GL's and simple groups not listed so far as basic building blocks.

4.

Decompose G into a direct product of irreducible factors.

5.

Recognize semidirect products G=N:H, where N is normal. Select a pair N, H with the following preferences:

1.

H is abelian

2.

N is abelian

2a.

N has many abelian invariants

3.

N is a direct product

3a.

N has many direct factors

4.

ϕ: H → Aut(N), h ↦ (n ↦ n^h) is injective.

6.

Fall back to non-splitting extensions: If the centre or the commutator factor group is non-trivial, write G as Z(G).G/Z(G) or G'.G/G', respectively. Otherwise if the Frattini subgroup is non-trivial, write G as Φ(G).G/Φ(G).

7.

If no decomposition is found (maybe this is not the case for any finite group), try to identify G in the perfect groups library. If this fails also, then return a string describing this situation.

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

1.

"Most" groups do not have "nice" decompositions. This is in some contrast to what is often taught in elementary courses on group theory, where it is sometimes suggested that basically every group can be written as iterated direct or semidirect product of cyclic groups and nonabelian simple groups.

2.

In particular many p-groups have very "similar" structure, and StructureDescription can only exhibit a little of it. Changing this would likely make the output not essentially easier to read than a pc presentation.

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(PSL(4,2));
"A8"

39.7 Cosets

39.7-1 RightCoset
‣ RightCoset( U, g )( operation )

returns the right coset of U with representative g, which is the set of all elements of the form ug for all u ∈ 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) ]

39.7-2 RightCosets
‣ RightCosets( G, U )( function )
‣ RightCosetsNC( G, U )( operation )

computes a duplicate free list of right cosets U g for g ∈ 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
‣ CanonicalRightCosetElement( U, g )( operation )

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
‣ IsRightCoset( obj )( category )

The category of right cosets.

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 CosetDecomposition
‣ CosetDecomposition( G, S )( function )

For a finite group G and a subgroup SleG 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
‣ RightTransversal( G, U )( operation )

A right transversal t is a list of representatives for the set UG 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 [G: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
‣ DoubleCoset( U, g, V )( operation )

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 ∈ U, v ∈ 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
‣ RepresentativesContainedRightCosets( D )( attribute )

A double coset 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 = ⋃_{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
‣ DoubleCosets( G, U, V )( operation )
‣ DoubleCosetsNC( G, U, V )( operation )

computes a duplicate free list of all double cosets U g V for g ∈ 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
‣ IsDoubleCoset( obj )( category )

The category of double cosets.

39.9-5 DoubleCosetRepsAndSizes
‣ DoubleCosetRepsAndSizes( G, U, V )( operation )

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
‣ InfoCoset( info class )

The information function for coset and double coset operations is InfoCoset.

39.10 Conjugacy Classes

39.10-1 ConjugacyClass
‣ ConjugacyClass( G, g )( operation )

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
‣ ConjugacyClasses( G )( attribute )

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).

For other groups the method of [Hul00] is employed.

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

random

The classes are computed by random search. See ConjugacyClassesByRandomSearch (39.10-3) below.

action

The classes are computed by action of G on itself. See ConjugacyClassesByOrbits (39.10-4) below.

noaction

Even for small groups ConjugacyClassesByOrbits (39.10-4) is not used as a default. This can be useful if the elements of the group use a lot of memory.

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 ConjugacyClassesGAP will try to select a suitable method on its own. The method specifications are provided here mainly for expert use.

39.10-3 ConjugacyClassesByRandomSearch
‣ ConjugacyClassesByRandomSearch( G )( function )

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
‣ ConjugacyClassesByOrbits( G )( function )

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
‣ NrConjugacyClasses( G )( attribute )

returns the number of conjugacy classes of G.

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

39.10-6 RationalClass
‣ RationalClass( G, g )( operation )

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
‣ RationalClasses( G )( attribute )

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
‣ GaloisGroup( ratcl )( attribute )

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 (15.3-3)), 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
‣ IsConjugate( G, x, y )( operation )
‣ IsConjugate( G, U, V )( operation )

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

RepresentativeAction (41.6-1) can be used to obtain conjugating elements.

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

39.10-10 NthRootsInGroup
‣ NthRootsInGroup( G, e, n )( function )

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
‣ Normalizer( G, U )( operation )
‣ Normalizer( G, g )( operation )

For two groups G, U, Normalizer computes the normalizer N_G(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_G(⟨ g ⟩).

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

39.11-2 Core
‣ Core( S, U )( operation )

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
‣ PCore( G, p )( function )

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

39.11-4 NormalClosure
‣ NormalClosure( G, U )( operation )

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

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

39.11-5 NormalIntersection
‣ NormalIntersection( G, U )( operation )

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
‣ ComplementClassesRepresentatives( G, N )( operation )

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 only methods for a solvable N are available.

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

39.11-7 InfoComplement
‣ InfoComplement( info class )

Info class for the complement routines.

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
‣ TrivialSubgroup( G )( attribute )
gap> TrivialSubgroup(g);
Group(())

39.12-2 CommutatorSubgroup
‣ CommutatorSubgroup( G, H )( operation )

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 ∈ G and h ∈ H, that is the group ⟨ [ g, h ] ∣ g ∈ G, h ∈ H.

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
‣ DerivedSubgroup( G )( attribute )

The derived subgroup 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 G/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), (1,4,3) ])

39.12-4 CommutatorLength
‣ CommutatorLength( G )( attribute )

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
‣ FittingSubgroup( G )( attribute )

The Fitting subgroup of a group G is its largest nilpotent normal subgroup.

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

39.12-6 FrattiniSubgroup
‣ FrattiniSubgroup( G )( attribute )

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

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

39.12-7 PrefrattiniSubgroup
‣ PrefrattiniSubgroup( G )( attribute )

returns a Prefrattini subgroup of the finite solvable group G.

A factor M/N of G is called a Frattini factor if M/N is contained in the Frattini subgroup of 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
‣ PerfectResiduum( G )( attribute )

is the smallest normal subgroup of G that has a solvable factor group.

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

39.12-9 RadicalGroup
‣ RadicalGroup( G )( attribute )

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

gap> RadicalGroup(SL(2,5));
<group of 2x2 matrices of size 2 over GF(5)>
gap> Size(last);
2

39.12-10 Socle
‣ Socle( G )( attribute )

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
‣ SupersolvableResiduum( G )( attribute )

is the supersolvable residuum of the group G, that is, its smallest normal subgroup N such that the factor group G / N is supersolvable.

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

39.12-12 PRump
‣ PRump( G, p )( function )

For a prime p, the p-rump of a group G is the subgroup G' G^p.

@example missing!@

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.

  1. For each prime p dividing the order of G, there exists a p-complement (see SylowComplement (39.13-2)).

  2. For each set P of primes dividing the order of G, there exists a P-Hall subgroup (see HallSubgroup (39.13-3)).

  3. G has a Sylow system (see SylowSystem (39.13-4)).

  4. G has a complement system (see ComplementSystem (39.13-5)).

39.13-1 SylowSubgroup
‣ SylowSubgroup( G, p )( function )

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
‣ SylowComplement( G, p )( function )

returns a Sylow p-complement of the finite group G. This is a subgroup U of order coprime to p such that the index [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
‣ HallSubgroup( G, P )( function )

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> HallSubgroup(h,[3,31]);
Group([ (2,29,7)(3,30,11)(4,28,8)(5,27,9)(6,31,10)(12,24,21)(13,25,20)
(14,23,17)(15,22,19)(16,26,18), (1,20,31,23,11,25,29,3,21,24,6,19,5,
17,12,14,4,18,8,10,30,27,16,26,22,13,7,15,28,9,2) ])
gap> HallSubgroup(h,[5,31]);
fail

39.13-4 SylowSystem
‣ SylowSystem( G )( attribute )

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
‣ ComplementSystem( G )( attribute )

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 |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
‣ HallSystem( G )( attribute )

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
‣ Omega( G, p[, n] )( operation )

For a p-group G, one defines Ω_n(G) = { g ∈ G ∣ g^{p^n} = 1 }. The default value for n is 1.

@At the moment methods exist only for abelian G and n=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
‣ Agemo( G, p[, n] )( function )

For a p-group G, one defines ℧_n(G) = ⟨ g^{p^n} ∣ g ∈ G. 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 (39.15-23) and IsPNilpotent (39.15-24), 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
‣ IsCyclic( G )( property )

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
‣ IsElementaryAbelian( G )( property )

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
‣ IsNilpotentGroup( G )( property )

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
‣ NilpotencyClassOfGroup( G )( attribute )

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));

If G is not nilpotent an error is issued.

39.15-5 IsPerfectGroup
‣ IsPerfectGroup( G )( property )

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

39.15-6 IsSolvableGroup
‣ IsSolvableGroup( G )( property )

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
‣ IsPolycyclicGroup( G )( property )

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
‣ IsSupersolvableGroup( G )( property )

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

39.15-9 IsMonomialGroup
‣ IsMonomialGroup( G )( property )

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

39.15-10 IsSimpleGroup
‣ IsSimpleGroup( G )( property )

A group is simple if it is nontrivial and has no nontrivial normal subgroups.

39.15-11 IsAlmostSimpleGroup
‣ IsAlmostSimpleGroup( G )( property )

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 IsomorphismTypeInfoFiniteSimpleGroup
‣ IsomorphismTypeInfoFiniteSimpleGroup( G )( attribute )
‣ IsomorphismTypeInfoFiniteSimpleGroup( n )( attribute )

For a finite simple group G, IsomorphismTypeInfoFiniteSimpleGroup returns a record with the components series, name and possibly parameter, describing the isomorphism type of G. The component name is a string that gives name(s) for G, and series is a string that 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.)

"A"

Alternating groups, parameter gives the natural degree.

"L"

Linear groups (Chevalley type A), parameter is a list [ n, q ] that indicates L(n,q).

"2A"

Twisted Chevalley type ^2A, parameter is a list [ n, q ] that indicates ^2A(n,q).

"B"

Chevalley type B, parameter is a list [n, q ] that indicates B(n,q).

"2B"

Twisted Chevalley type ^2B, parameter is a value q that indicates ^2B(2,q).

"C"

Chevalley type C, parameter is a list [ n, q ] that indicates C(n,q).

"D"

Chevalley type D, parameter is a list [ n, q ] that indicates D(n,q).

"2D"

Twisted Chevalley type ^2D, parameter is a list [ n, q ] that indicates ^2D(n,q).

"3D"

Twisted Chevalley type ^3D, parameter is a value q that indicates ^3D(4,q).

"E"

Exceptional Chevalley type E, parameter is a list [ n, q ] that indicates E_n(q). The value of n is 6, 7, or 8.

"2E"

Twisted exceptional Chevalley type E_6, parameter is a value q that indicates ^2E_6(q).

"F"

Exceptional Chevalley type F, parameter is a value q that indicates F(4,q).

"2F"

Twisted exceptional Chevalley type ^2F (Ree groups), parameter is a value q that indicates ^2F(4,q).

"G"

Exceptional Chevalley type G, parameter is a value q that indicates G(2,q).

"2G"

Twisted exceptional Chevalley type ^2G (Ree groups), parameter is a value q that indicates ^2G(2,q).

"Spor"

Sporadic simple groups, name gives the name.

"Z"

Cyclic groups of prime size, parameter gives the size.

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" )

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" )
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 ] )

39.15-13 SimpleGroup
‣ SimpleGroup( id[, param] )( function )

This function will construct an instance of the specified 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 who do not have a permutation representation of small degree the ATLASREP package might need to be installed to construct theses 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)

39.15-14 SimpleGroupsIterator
‣ SimpleGroupsIterator( [start[, end]] )( function )

This function returns an iterator that will run over all simple groups, starting at order start if specified, up to order 10^18 (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 ]

39.15-15 SmallSimpleGroup
‣ SmallSimpleGroup( order[, i] )( function )

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)

39.15-16 AllSmallNonabelianSimpleGroups
‣ AllSmallNonabelianSimpleGroups( orders )( function )

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)" ]

39.15-17 IsFinitelyGeneratedGroup
‣ IsFinitelyGeneratedGroup( G )( property )

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.

39.15-18 IsSubsetLocallyFiniteGroup
‣ IsSubsetLocallyFiniteGroup( U )( property )

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.

39.15-19 IsPGroup
‣ IsPGroup( G )( property )

A p-group is a finite group whose order (see Size (30.4-6)) 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.

39.15-20 PrimePGroup
‣ PrimePGroup( G )( attribute )

If G is a nontrivial p-group (see IsPGroup (39.15-19)), 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.)

39.15-21 PClassPGroup
‣ PClassPGroup( G )( attribute )

The p-class of a p-group G (see IsPGroup (39.15-19)) 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.

39.15-22 RankPGroup
‣ RankPGroup( G )( attribute )

For a p-group G (see IsPGroup (39.15-19)), 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

39.15-23 IsPSolvable
‣ IsPSolvable( G, p )( function )

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

39.15-24 IsPNilpotent
‣ IsPNilpotent( G, p )( function )

A group is p-nilpotent if it possesses a normal p-complement.

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
‣ AbelianInvariants( G )( attribute )

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 G/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
‣ Exponent( G )( attribute )

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 ∈ G.

gap> Exponent(g);
12

39.16-3 EulerianFunction
‣ EulerianFunction( G, n )( operation )

returns the number of n-tuples (g_1, g_2, ..., 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 ϕ_n(G) is used for the value returned by EulerianFunction, and the quotient of ϕ_n(G) by the order of the automorphism group of G is called d_n(G). If G is a nonabelian simple group then d_n(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 > ⋯ > U_m = ⟨ 1 ⟩ of subgroups considered. A series also may stop without reaching G or ⟨ 1 ⟩.

A series is called subnormal if every U_{i+1} is normal in U_i.

A series is called normal if every U_i is normal in G.

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
‣ ChiefSeries( G )( attribute )

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
‣ ChiefSeriesThrough( G, l )( operation )

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
‣ ChiefSeriesUnderAction( H, G )( operation )

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
‣ SubnormalSeries( G, U )( operation )

If U is a subgroup of G this operation returns a subnormal series that descends from G to a subnormal subgroup V ≥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) ]) ]

39.17-5 CompositionSeries
‣ CompositionSeries( G )( attribute )

A composition series is a subnormal series which cannot be refined. This attribute returns one composition series (of potentially many possibilities).

39.17-6 DisplayCompositionSeries
‣ DisplayCompositionSeries( G )( function )

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)
 | Z(2)
S (5 gens, size 2520)
 | A(7)
1 (0 gens, size 1)

39.17-7 DerivedSeriesOfGroup
‣ DerivedSeriesOfGroup( G )( attribute )

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
‣ DerivedLength( G )( attribute )

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
‣ ElementaryAbelianSeries( G )( attribute )
‣ ElementaryAbelianSeriesLargeSteps( G )( attribute )
‣ ElementaryAbelianSeries( list )( attribute )

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
‣ InvariantElementaryAbelianSeries( G, morph[, N[, fine]] )( operation )

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
‣ LowerCentralSeriesOfGroup( G )( attribute )

The lower central series of a group G is defined as U_{i+1}:= [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
‣ UpperCentralSeriesOfGroup( G )( attribute )

The upper central series of a group G is defined as an ending series U_i / U_{i+1}:= Z(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
‣ PCentralSeries( G, p )( function )

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

39.17-14 JenningsSeries
‣ JenningsSeries( G )( attribute )

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

39.17-15 DimensionsLoewyFactors
‣ DimensionsLoewyFactors( G )( attribute )

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
‣ AscendingChain( G, U )( function )

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
‣ IntermediateGroup( G, U )( function )

This routine tries to find a subgroup E of G, such that G > E > 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
‣ IntermediateSubgroups( G, U )( operation )

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.

39.18 Factor Groups

39.18-1 NaturalHomomorphismByNormalSubgroup
‣ NaturalHomomorphismByNormalSubgroup( G, N )( function )
‣ NaturalHomomorphismByNormalSubgroupNC( G, N )( function )

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 G/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
‣ FactorGroup( G, N )( function )
‣ FactorGroupNC( G, N )( operation )

returns the image of the NaturalHomomorphismByNormalSubgroup(G,N). The homomorphism will be stored in the attribute NaturalHomomorphism of the result. 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> FactorGroup(g,n);
Group([ f1, f2 ])

39.18-3 CommutatorFactorGroup
‣ CommutatorFactorGroup( G )( attribute )

computes the commutator factor group G/G' of the group G.

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

39.18-4 MaximalAbelianQuotient
‣ MaximalAbelianQuotient( G )( attribute )

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
‣ HasAbelianFactorGroup( G, N )( operation )

tests whether G / N is abelian (without explicitly constructing the factor group).

39.18-6 HasElementaryAbelianFactorGroup
‣ HasElementaryAbelianFactorGroup( G, N )( operation )

tests whether G / N is elementary abelian (without explicitly constructing the factor group).

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

39.18-7 CentralizerModulo
‣ CentralizerModulo( G, N, elm )( operation )

Computes the full preimage of the centralizer C_{G/N}(elmN) 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
‣ ConjugacyClassSubgroups( G, U )( operation )

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
‣ IsConjugacyClassSubgroupsRep( obj )( representation )
‣ IsConjugacyClassSubgroupsByStabilizerRep( obj )( representation )

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
‣ ConjugacyClassesSubgroups( G )( attribute )

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
‣ ConjugacyClassesMaximalSubgroups( G )( attribute )

returns the conjugacy classes of maximal subgroups of G. Representatives of the classes can be computed directly by MaximalSubgroupClassReps (39.19-6).

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

39.19-5 AllSubgroups
‣ AllSubgroups( G )( function )

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) ]) ]

39.19-6 MaximalSubgroupClassReps
‣ MaximalSubgroupClassReps( G )( attribute )

returns a list of conjugacy representatives of the maximal subgroups of G.

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

39.19-7 MaximalSubgroups
‣ MaximalSubgroups( G )( attribute )

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) ]) ]

39.19-8 NormalSubgroups
‣ NormalSubgroups( G )( attribute )

returns a list of all normal subgroups of G.

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

The algorithm for the computation of normal subgroups is described in [Hul98].

39.19-9 MaximalNormalSubgroups
‣ MaximalNormalSubgroups( G )( attribute )

is a list containing those proper normal subgroups of the group G that are maximal among the proper normal subgroups.

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

39.19-10 MinimalNormalSubgroups
‣ MinimalNormalSubgroups( G )( attribute )

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

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

39.20 Subgroup Lattice

39.20-1 LatticeSubgroups
‣ LatticeSubgroups( G )( attribute )

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
‣ ClassElementLattice( C, n )( operation )

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
‣ DotFileLatticeSubgroups( L, file )( function )

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 http://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
‣ MaximalSubgroupsLattice( lat )( attribute )

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
‣ MinimalSupergroupsLattice( lat )( attribute )

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 RepresentativesPerfectSubgroups
‣ RepresentativesPerfectSubgroups( G )( attribute )
‣ RepresentativesSimpleSubgroups( G )( attribute )

returns a list of conjugacy representatives of perfect (respectively simple) subgroups of G. This uses the library of perfect groups (see PerfectGroup (50.8-2)), 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);
[ Group([ (3,9,11)(4,8,6)(5,10,7), (1,3)(2,11)(4,6)(7,8) ]), 
  Group([ (2,3,4)(5,6,8)(7,11,9), (1,3)(2,11)(4,6)(7,8) ]), 
  Group([ (4,11,10,9)(5,7,8,6), (1,6,4)(2,9,7)(8,11,10) ]), 
  Group([ (3,5,8)(4,11,7)(6,9,10), (1,3)(2,11)(4,6)(7,8) ]), M(11), 
  Group(()) ]
gap> List(r,Size);
[ 60, 60, 360, 660, 7920, 1 ]

39.20-7 ConjugacyClassesPerfectSubgroups
‣ ConjugacyClassesPerfectSubgroups( G )( attribute )

returns a list of the conjugacy classes of perfect subgroups of G. (see RepresentativesPerfectSubgroups (39.20-6).)

gap> ConjugacyClassesPerfectSubgroups(m11);
[ Group( [ ( 3, 9,11)( 4, 8, 6)( 5,10, 7), 
      ( 1, 3)( 2,11)( 4, 6)( 7, 8) ] )^G, 
  Group( [ ( 2, 3, 4)( 5, 6, 8)( 7,11, 9), 
      ( 1, 3)( 2,11)( 4, 6)( 7, 8) ] )^G, 
  Group( [ ( 4,11,10, 9)( 5, 7, 8, 6), ( 1, 6, 4)( 2, 9, 7)( 8,11,10) 
     ] )^G, 
  Group( [ ( 3, 5, 8)( 4,11, 7)( 6, 9,10), 
      ( 1, 3)( 2,11)( 4, 6)( 7, 8) ] )^G, M(11)^G, Group( () )^G ]

39.20-8 Zuppos
‣ Zuppos( G )( attribute )

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.

39.20-9 InfoLattice
‣ InfoLattice( info class )

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
‣ LatticeByCyclicExtension( G[, func[, noperf]] )( function )

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
‣ InvariantSubgroupsElementaryAbelianGroup( G, homs[, dims] )( function )

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(()), Group([ (1,2,3)(4,5,6)(7,8,9) ]), 
  Group([ (1,3,2)(7,8,9), (1,3,2)(4,5,6) ]), 
  Group([ (7,8,9), (4,5,6), (1,2,3) ]) ]

39.21-3 SubgroupsSolvableGroup
‣ SubgroupsSolvableGroup( G[, opt] )( function )

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 put restrictions on the subgroups computed. The following record components of opt are recognized and have the following effects:

actions

must be a list of automorphisms of G. If given, only groups which are invariant under all these automorphisms are computed. The algorithm must know the normalizer in G of the group generated by actions (defined formally by embedding in the semidirect product of G with actions). This can be given in the component funcnorm and will be computed if this component is not given.

normal

if set to true only normal subgroups are guaranteed to be returned (though some of the returned subgroups might still be not normal).

consider

a function to restrict the groups computed. This must be a function of five parameters, C, A, N, B, M, that are interpreted as follows: The arguments are subgroups of a factor F of G in the relation F ≥ C > A > N > B > M. N and M are normal subgroups. C is the full preimage of the normalizer of A/N in F/N. When computing modulo M and looking for subgroups U such that U ∩ N = B and ⟨ U, N ⟩ = A, this function is called. If it returns false then all potential groups U (and therefore all groups later arising from them) are disregarded. This can be used for example to compute only subgroups of certain sizes.

(This is just a restriction to speed up computations. The function may still return (invariant) subgroups which don't fulfill this condition!) This parameter is used to permit calculations of some subgroups if the set of all subgroups would be too large to handle.

The actual groups C, A, N and B which are passed to this function are not necessarily subgroups of G but might be subgroups of a proper factor group F = G/H. Therefore the consider function may not relate the parameter groups to G.

retnorm

if set to true the function not only returns a list subs of subgroups but also a corresponding list norms of normalizers in the form [ subs, norms ].

series

is an elementary abelian series of G which will be used for the computation.

groups

is a list of groups to seed the calculation. Only subgroups of these groups are constructed.

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> List(l,Size);
[ 1, 3, 9, 27, 54, 2, 6, 18, 108, 4, 216, 8 ]
gap> Length(ConjugacyClassesSubgroups(g)); # to compare
162

39.21-4 SizeConsiderFunction
‣ SizeConsiderFunction( size )( function )

This function returns a function consider of four 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> List(l,Size);
[ 1, 3, 9, 27, 54, 6, 18, 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
‣ ExactSizeConsiderFunction( size )( function )

This function returns a function consider of four 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> List(l,Size);
[ 1, 3, 9, 27, 54, 6, 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
‣ InfoPcSubgroup( info class )

Information function for the subgroup lattice functions using pcgs.

39.22 Special Generating Sets

39.22-1 GeneratorsSmallest
‣ GeneratorsSmallest( G )( attribute )

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 = ⟨ l ⟩ and l_i not ∈ ⟨ l_1, ..., l_{i-1} ⟩ (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
‣ LargestElementGroup( G )( attribute )

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

39.22-3 MinimalGeneratingSet
‣ MinimalGeneratingSet( G )( attribute )

returns a generating set of G of minimal possible length.

Note that –apart from special cases– currently there are only efficient methods known to compute minimal generating sets of finite solvable groups and of finitely generated nilpotent groups. Hence so far these are the only cases for which methods are available. The former case is covered by a method implemented in the GAP library, while the second case requires the package Polycyclic.

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
‣ SmallGeneratingSet( G )( attribute )

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
‣ IndependentGeneratorsOfAbelianGroup( A )( attribute )

returns a list of generators a_1, a_2, ... 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
‣ IndependentGeneratorExponents( G, g )( operation )

For an abelian group G, with IndependentGeneratorsOfAbelianGroup (39.22-5) value the list [ a_1, ..., a_n ], this operation returns the exponent vector [ e_1, ..., e_n ] to represent g = ∏_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) = { γ: G/M → M ∣ ∀ g_1, g_2 ∈ G : γ(g_1 M ⋅ g_2 M ) = γ(g_1 M)^{g_2} ⋅ γ(g_2 M) }

and is a GF(p)-vector space.

The group of 1-coboundaries B^1( G/M, M ) is defined as

B^1(G/M, M) = { γ : G/M → M ∣ ∃ m ∈ M ∀ g ∈ G : γ(gM) = (m^{-1})^g ⋅ m }

It also is a GF(p)-vector space.

Let α be the isomorphism of M into a row vector space cal W and (g_1, ..., g_l) representatives for a generating set of G/M. Then there exists a monomorphism β of Z^1( G/M, M ) in the l-fold direct sum of cal W, such that β( γ ) = ( α( γ(g_1 M) ),..., α( γ(g_l M) ) ) for every γ ∈ Z^1( G/M, M ).

39.23-1 OneCocycles
‣ OneCocycles( G, M )( operation )
‣ OneCocycles( G, mpcgs )( operation )
‣ OneCocycles( gens, M )( operation )
‣ OneCocycles( gens, mpcgs )( operation )

Computes the group of 1-cocycles Z^1(G/M,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
‣ OneCoboundaries( G, M )( operation )

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:

generators

Is a list of representatives for a generating set of G/M. Cocycles are represented with respect to these generators.

oneCocycles

A space of row vectors over GF(p), representing Z^1. The vectors are represented in dimension a ⋅ b where a is the length of generators and p^b the size of M.

oneCoboundaries

A space of row vectors that represents B^1.

cocycleToList

is a function to convert a cocycle (a row vector in oneCocycles) to a corresponding list of elements of M.

listToCocycle

is a function to convert a list of elements of M to a cocycle.

isSplitExtension

indicates whether G splits over M. The following components are only bound if the extension splits. Note that if M is given by a modulo pcgs all subgroups are given as subgroups of G by generators corresponding to generators and thus may not contain the denominator of the modulo pcgs. In this case taking the closure with this denominator will give the full preimage of the complement in the factor group.

complement

One complement to M in G.

cocycleToComplement( cyc )

is a function that takes a cocycle from oneCocycles and returns the corresponding complement to M in G (with respect to the fixed complement complement).

complementToCocycle(U)

is a function that takes a complement and returns the corresponding cocycle.

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(G/M, M) = Z^1(G/M, M) / B^1(G/M, 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
‣ OCOneCocycles( ocr, onlySplit )( operation )

is the more technical function to compute 1-cocycles. It takes an 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
‣ ComplementClassesRepresentativesEA( G, N )( operation )

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
‣ InfoCoh( info class )

The info class for the cohomology calculations is 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
‣ EpimorphismSchurCover( G[, pl] )( operation )

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
‣ SchurCover( G )( operation )

returns one (of possibly several) Schur covers of the group G.

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 ] -> [ (3,4), (2,4,3), (1,3)(2,4) ]
gap> Size(Source(epi));
48

If the group becomes bigger, Schur Cover calculations might become unfeasible.

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
‣ AbelianInvariantsMultiplier( G )( attribute )

returns a list of the abelian invariants of the Schur multiplier of G.

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 ]

(Note that the last command from the example will take some time.)

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
‣ Epicentre( G )( attribute )
‣ ExteriorCentre( G )( attribute )

There are various ways of describing the epicentre of a group G. It is the smallest normal subgroup N of G such that 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
‣ NonabelianExteriorSquare( G )( operation )

Computes the nonabelian exterior square GG 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
‣ EpimorphismNonabelianExteriorSquare( G )( operation )

Computes the mapping GGG. The kernel of this mapping is equal to the Schur multiplier of G.

39.24-7 IsCentralFactor
‣ IsCentralFactor( G )( property )

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
‣ BasicSpinRepresentationOfSymmetricGroup( n, p, sign )( function )

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 p ≥ 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.

39.24-10 SchurCoverOfSymmetricGroup
‣ SchurCoverOfSymmetricGroup( n, p, sign )( operation )

Constructs a Schur cover of SymmetricGroup(n) as a faithful, irreducible matrix group in characteristic p (p ≠ 2). For n ≥ 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. For n ≤ 3, the symmetric group is its own Schur cover and sign is ignored. For p = 2, there is no faithful, irreducible representation of the Schur cover unless n = 1 or n = 3, so fail is returned if p = 2. For p = 3, n = 3, the representation is indecomposable, but reducible. The field of the matrix group is generally GF(p^2) if p > 0, and an abelian number field if p = 0.

39.24-11 DoubleCoverOfAlternatingGroup
‣ DoubleCoverOfAlternatingGroup( n, p )( operation )

Constructs a double cover of AlternatingGroup(n) as a faithful, completely reducible matrix group in characteristic p (p ≠ 2) for n ≥ 4. For n ≥ 3, the symmetric group is its own Schur cover so 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 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).

39.25-1 CanEasilyTestMembership
‣ CanEasilyTestMembership( G )( function )

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.

39.25-2 CanEasilyComputeWithIndependentGensAbelianGroup
‣ CanEasilyComputeWithIndependentGensAbelianGroup( G )( function )

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.

39.25-3 CanComputeSize
‣ CanComputeSize( dom )( function )

This filter indicates whether the size of the domain dom (which might be infinity (18.2-1)) can be computed.

39.25-4 CanComputeSizeAnySubgroup
‣ CanComputeSizeAnySubgroup( G )( function )

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.)

39.25-5 CanComputeIndex
‣ CanComputeIndex( G, H )( function )

This function indicates whether the index [G:H] (which might be infinity (18.2-1)) can be computed. It assumes that HG (see CanComputeIsSubset (39.25-6)).

39.25-6 CanComputeIsSubset
‣ CanComputeIsSubset( A, B )( operation )

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

39.25-7 KnowsHowToDecompose
‣ KnowsHowToDecompose( G[, gens] )( property )

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.

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