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39 Groups

39.15 Group Properties

39.15-1 IsCyclic

39.15-2 IsElementaryAbelian

39.15-3 IsNilpotentGroup

39.15-4 NilpotencyClassOfGroup

39.15-5 IsPerfectGroup

39.15-6 IsSolvableGroup

39.15-7 IsPolycyclicGroup

39.15-8 IsSupersolvableGroup

39.15-9 IsMonomialGroup

39.15-10 IsSimpleGroup

39.15-11 IsAlmostSimpleGroup

39.15-12 IsQuasisimpleGroup

39.15-13 IsomorphismTypeInfoFiniteSimpleGroup

39.15-14 SimpleGroup

39.15-15 SimpleGroupsIterator

39.15-16 SmallSimpleGroup

39.15-17 AllSmallNonabelianSimpleGroups

39.15-18 IsFinitelyGeneratedGroup

39.15-19 IsSubsetLocallyFiniteGroup

39.15-20 IsPGroup

39.15-21 IsPowerfulPGroup

39.15-22 PrimePGroup

39.15-23 PClassPGroup

39.15-24 RankPGroup

39.15-25 IsPSolvable

39.15-26 IsPNilpotent

39.15-1 IsCyclic

39.15-2 IsElementaryAbelian

39.15-3 IsNilpotentGroup

39.15-4 NilpotencyClassOfGroup

39.15-5 IsPerfectGroup

39.15-6 IsSolvableGroup

39.15-7 IsPolycyclicGroup

39.15-8 IsSupersolvableGroup

39.15-9 IsMonomialGroup

39.15-10 IsSimpleGroup

39.15-11 IsAlmostSimpleGroup

39.15-12 IsQuasisimpleGroup

39.15-13 IsomorphismTypeInfoFiniteSimpleGroup

39.15-14 SimpleGroup

39.15-15 SimpleGroupsIterator

39.15-16 SmallSimpleGroup

39.15-17 AllSmallNonabelianSimpleGroups

39.15-18 IsFinitelyGeneratedGroup

39.15-19 IsSubsetLocallyFiniteGroup

39.15-20 IsPGroup

39.15-21 IsPowerfulPGroup

39.15-22 PrimePGroup

39.15-23 PClassPGroup

39.15-24 RankPGroup

39.15-25 IsPSolvable

39.15-26 IsPNilpotent

39.17 Subgroup Series

39.17-1 ChiefSeries

39.17-2 ChiefSeriesThrough

39.17-3 ChiefSeriesUnderAction

39.17-4 SubnormalSeries

39.17-5 CompositionSeries

39.17-6 DisplayCompositionSeries

39.17-7 DerivedSeriesOfGroup

39.17-8 DerivedLength

39.17-9 ElementaryAbelianSeries

39.17-10 InvariantElementaryAbelianSeries

39.17-11 LowerCentralSeriesOfGroup

39.17-12 UpperCentralSeriesOfGroup

39.17-13 PCentralSeries

39.17-14 JenningsSeries

39.17-15 DimensionsLoewyFactors

39.17-16 AscendingChain

39.17-17 IntermediateGroup

39.17-18 IntermediateSubgroups

39.17-19 StructuralSeriesOfGroup

39.17-1 ChiefSeries

39.17-2 ChiefSeriesThrough

39.17-3 ChiefSeriesUnderAction

39.17-4 SubnormalSeries

39.17-5 CompositionSeries

39.17-6 DisplayCompositionSeries

39.17-7 DerivedSeriesOfGroup

39.17-8 DerivedLength

39.17-9 ElementaryAbelianSeries

39.17-10 InvariantElementaryAbelianSeries

39.17-11 LowerCentralSeriesOfGroup

39.17-12 UpperCentralSeriesOfGroup

39.17-13 PCentralSeries

39.17-14 JenningsSeries

39.17-15 DimensionsLoewyFactors

39.17-16 AscendingChain

39.17-17 IntermediateGroup

39.17-18 IntermediateSubgroups

39.17-19 StructuralSeriesOfGroup

39.19 Sets of Subgroups

39.19-1 ConjugacyClassSubgroups

39.19-2 IsConjugacyClassSubgroupsRep

39.19-3 ConjugacyClassesSubgroups

39.19-4 ConjugacyClassesMaximalSubgroups

39.19-5 MaximalSubgroupClassReps

39.19-6 LowIndexSubgroups

39.19-7 AllSubgroups

39.19-8 MaximalSubgroups

39.19-9 NormalSubgroups

39.19-10 MaximalNormalSubgroups

39.19-11 MinimalNormalSubgroups

39.19-12 CharacteristicSubgroups

39.19-1 ConjugacyClassSubgroups

39.19-2 IsConjugacyClassSubgroupsRep

39.19-3 ConjugacyClassesSubgroups

39.19-4 ConjugacyClassesMaximalSubgroups

39.19-5 MaximalSubgroupClassReps

39.19-6 LowIndexSubgroups

39.19-7 AllSubgroups

39.19-8 MaximalSubgroups

39.19-9 NormalSubgroups

39.19-10 MaximalNormalSubgroups

39.19-11 MinimalNormalSubgroups

39.19-12 CharacteristicSubgroups

39.20 Subgroup Lattice

39.20-1 LatticeSubgroups

39.20-2 ClassElementLattice

39.20-3 DotFileLatticeSubgroups

39.20-4 MaximalSubgroupsLattice

39.20-5 MinimalSupergroupsLattice

39.20-6 LowLayerSubgroups

39.20-7 ContainedConjugates

39.20-8 ContainingConjugates

39.20-9 MinimalFaithfulPermutationDegree

39.20-10 RepresentativesPerfectSubgroups

39.20-11 ConjugacyClassesPerfectSubgroups

39.20-12 Zuppos

39.20-13 InfoLattice

39.20-1 LatticeSubgroups

39.20-2 ClassElementLattice

39.20-3 DotFileLatticeSubgroups

39.20-4 MaximalSubgroupsLattice

39.20-5 MinimalSupergroupsLattice

39.20-6 LowLayerSubgroups

39.20-7 ContainedConjugates

39.20-8 ContainingConjugates

39.20-9 MinimalFaithfulPermutationDegree

39.20-10 RepresentativesPerfectSubgroups

39.20-11 ConjugacyClassesPerfectSubgroups

39.20-12 Zuppos

39.20-13 InfoLattice

39.24 Schur Covers and Multipliers

39.24-1 EpimorphismSchurCover

39.24-2 SchurCover

39.24-3 AbelianInvariantsMultiplier

39.24-4 Epicentre

39.24-5 NonabelianExteriorSquare

39.24-6 EpimorphismNonabelianExteriorSquare

39.24-7 IsCentralFactor

39.24-8 Covering groups of symmetric groups

39.24-9 BasicSpinRepresentationOfSymmetricGroup

39.24-10 SchurCoverOfSymmetricGroup

39.24-11 DoubleCoverOfAlternatingGroup

39.24-1 EpimorphismSchurCover

39.24-2 SchurCover

39.24-3 AbelianInvariantsMultiplier

39.24-4 Epicentre

39.24-5 NonabelianExteriorSquare

39.24-6 EpimorphismNonabelianExteriorSquare

39.24-7 IsCentralFactor

39.24-8 Covering groups of symmetric groups

39.24-9 BasicSpinRepresentationOfSymmetricGroup

39.24-10 SchurCoverOfSymmetricGroup

39.24-11 DoubleCoverOfAlternatingGroup

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

If not otherwise specified, in all examples in this chapter the group `g`

will be the symmetric group \(S_4\) acting on the letters \(\{ 1, \ldots, 4 \}\).

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 \(x^{{-1}} y\).

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).

`‣ Group` ( gen, ... ) | ( function ) |

`‣ Group` ( gens[, id] ) | ( function ) |

`Group( `

is the group generated by the arguments `gen`, ... )`gen`, ...

If the only argument `gens` is a list that is not a matrix then `Group( `

is the group generated by the elements of that list.`gens` )

If there are two arguments, a list `gens` and an element `id`, then `Group( `

is the group generated by the elements of `gens`, `id` )`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) ])

`‣ 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).

`‣ 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`.

`‣ 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 \(i\),

returns the \(i\)-th element of the list returned by `G`.i`GeneratorsOfGroup`

. Moreover, if `G` is a free group, and `name`

is the name of a generator of `G` then

also returns this generator.`G`.name

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.

`‣ 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) ])

`‣ 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) ])

`‣ 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

`‣ InfoGroup` | ( info class ) |

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

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.

`‣ 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) ])

`‣ Index` ( G, U ) | ( operation ) |

`‣ IndexNC` ( G, U ) | ( operation ) |

For a subgroup `U` of the group `G`, `Index`

returns the index \([\textit{G}:\textit{U}] = |\textit{G}| / |\textit{U}|\) of `U` in `G`. The `NC`

version does not test whether `U` is contained in `G`.

gap> Index(g,u); 4

`‣ 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

`‣ 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) ])

`‣ 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( `

and `U` )`IsSubset( `

; note that special methods for `G`, `U` )`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

`‣ 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 \(g \in \textit{G}\) and \(u \in \textit{U}\) the element \(u^g\) is a member of `U`. Note that `U` need not be a subgroup of `G`.

gap> IsNormal(g,u); false

`‣ IsCharacteristicSubgroup` ( G, N ) | ( operation ) |

tests whether `N` is invariant under all automorphisms of `G`.

gap> IsCharacteristicSubgroup(g,u); false

`‣ 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 `^`

.

`‣ ConjugateSubgroups` ( G, U ) | ( operation ) |

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

`‣ 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

`‣ 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.

`‣ 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>

`‣ 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 ])

`‣ 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`.

`‣ 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).

`‣ 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.

Using homomorphisms (see chapter 40) it is possible to express group elements as words in given generators: Create a free group (see `FreeGroup`

(37.2-1)) on the correct number of generators and create a homomorphism from this free group onto the group `G` in whose generators you want to factorize. Then the preimage of an element of `G` is a word in the free generators, that will map on this element again.

`‣ 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

`‣ 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) ] ]

`‣ GrowthFunctionOfGroup` ( G ) | ( attribute ) |

`‣ 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 ]

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

Note that the `StructureDescription`

is only *one* possible way of building up the given group from smaller pieces.

The option "short" is recognized - if this option is set, an abbreviated output format is used (e.g. `"6x3"`

instead of `"C6 x C3"`

).

If the `Name`

(12.8-2) attribute is not bound, but `StructureDescription`

is, `View`

(6.3-3) prints the value of the attribute `StructureDescription`

. The `Print`

(6.3-4)ed representation of a group is not affected by computing a `StructureDescription`

.

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 or 96.**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"`

. For infinite abelian groups,`"Z"`

denotes the group of integers.**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.**if

`G`is defined as a semidirect product of \(N\), \(H\) then select \(N\), \(H\),**2.**if

`G`is solvable, then select a solvable normal Hall subgroup \(N\), if exists, and consider the semidirect decomposition of \(N\) and \(\textit{G}/N\),**3.**find any nontrivial normal subgroup \(N\) which has a complement \(H\).

The option "nice" is recognized. If this option is set, then all semidirect products are computed in order to find a possibly nicer presentation. Note, that this may take a very long time if

`G`has many normal subgroups, e.g. if \(\textit{G}/\textit{G}'\) has many cyclic factors. If the option "nice" is set, then GAP would 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.**\(\phi: H \rightarrow\) Aut(\(N\)), \(h \mapsto (n \mapsto n^h)\) is injective.

**6.**Fall back to non-splitting extensions: If the centre or the commutator factor group is non-trivial, write

`G`as \(Z(\textit{G})\).\(\textit{G}/Z(\textit{G})\) or \(\textit{G}'\).\(\textit{G}/\textit{G}'\), respectively. Otherwise if the Frattini subgroup is non-trivial, write`G`as \(\Phi\)(`G`).`G`/\(\Phi\)(`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(SmallGroup(504,7)); "C7 : (C9 x Q8)" gap> StructureDescription(SmallGroup(504,7):nice); "(C7 : Q8) : C9" gap> StructureDescription(AbelianGroup([0,2,3])); "Z x C6" gap> StructureDescription(AbelianGroup([0,0,0,2,3,6]):short); "Z^3x6^2" gap> StructureDescription(PSL(4,2)); "A8"

`‣ 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 \(u \in \textit{U}\). `g` must be an element of a larger group `G` which contains `U`. For element operations such as `in`

a right coset behaves like a set of group elements.

Right cosets are external orbits for the action of `U` which acts via `OnLeftInverse`

(41.2-3). Of course the action of a larger group `G` on right cosets is via `OnRight`

(41.2-2).

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

`‣ RightCosets` ( G, U ) | ( function ) |

`‣ RightCosetsNC` ( G, U ) | ( operation ) |

computes a duplicate free list of right cosets `U` \(g\) for \(g \in\) `G`. A set of representatives for the elements in this list forms a right transversal of `U` in `G`. (By inverting the representatives one obtains a list of representatives of the left cosets of `U`.) The `NC`

version does not check whether `U` is a subgroup of `G`.

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

`‣ 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)

`‣ 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.

`‣ IsBiCoset` ( C ) | ( property ) |

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

`‣ CosetDecomposition` ( G, S ) | ( function ) |

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

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

`‣ RightTransversal` ( G, U ) | ( operation ) |

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

The object returned by `RightTransversal`

is not a plain list, but an object that behaves like an immutable list of length \([\textit{G}:\textit{U}]\), except if `U` is the trivial subgroup of `G` in which case `RightTransversal`

may return the sorted plain list of coset representatives.

The operation `PositionCanonical`

(21.16-3), called for a transversal \(t\) and an element \(g\) of `G`, will return the position of the representative in \(t\) that lies in the same coset of `U` as the element \(g\) does. (In comparison, `Position`

(21.16-1) will return `fail`

if the element is not equal to the representative.) Functions that implement group actions such as `Action`

(41.7-2) or `Permutation`

(41.9-1) (see Chapter 41) use `PositionCanonical`

(21.16-3), therefore it is possible to "act" on a right transversal to implement the action on the cosets. This is often much more efficient than acting on cosets.

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

Note that the elements of a right transversal are not necessarily "canonical" in the sense of `CanonicalRightCosetElement`

(39.7-3), but we may compute a list of canonical coset representatives by calling that function. (See also `PositionCanonical`

(21.16-3).)

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

`‣ 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 \(u \in \textit{U}\), \(v \in \textit{V}\). For element operations such as `in`

, a double coset behaves like a set of group elements. The double coset stores `U` in the attribute `LeftActingGroup`

, `g` as `Representative`

(30.4-7), and `V` as `RightActingGroup`

.

`‣ RepresentativesContainedRightCosets` ( D ) | ( attribute ) |

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

(39.7-3)) and form a set.

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

`‣ DoubleCosets` ( G, U, V ) | ( function ) |

`‣ DoubleCosetsNC` ( G, U, V ) | ( operation ) |

computes a duplicate free list of all double cosets `U` \(g\) `V` for \(g \in \textit{G}\). The groups `U` and `V` must be subgroups of the group `G`. The `NC`

version does not check whether `U` and `V` are subgroups of `G`.

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

`‣ IsDoubleCoset` ( obj ) | ( category ) |

The category of double cosets.

`‣ 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 ] ]

`‣ InfoCoset` | ( info class ) |

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

.

`‣ 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).)

`‣ 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 `ConjugacyClasses`

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

`‣ 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).

`‣ 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

`‣ NrConjugacyClasses` ( G ) | ( attribute ) |

returns the number of conjugacy classes of `G`.

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

`‣ 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.

`‣ 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) ) ]

`‣ 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 `

, the unit group of a residue class ring. The Galois group of a rational class `n` )`ratcl` is stored in the attribute `GaloisGroup`

as a subgroup of this group.

`‣ 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)

`‣ 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) ]

For the operations `Centralizer`

(35.4-4) and `Centre`

(35.4-5), see Chapter 35.

`‣ Normalizer` ( G, U ) | ( operation ) |

`‣ Normalizer` ( G, g ) | ( operation ) |

For two groups `G`, `U`, `Normalizer`

computes the normalizer \(N_{\textit{G}}(\textit{U})\), that is, the stabilizer of `U` under the conjugation action of `G`.

For a group `G` and a group element `g`, `Normalizer`

computes \(N_{\textit{G}}(\langle \textit{g} \rangle)\).

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

`‣ 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(())

`‣ PCore` ( G, p ) | ( operation ) |

The * p-core* of

`Core`

(39.11-2).gap> g:=DicyclicGroup(12);; gap> PCore(g,2); Group([ y3 ]) gap> PCore(g,2) = Core(g,SylowSubgroup(g,2)); true gap> PCore(g,3); Group([ y*y3 ]) gap> PCore(g,5); Group([ ]) gap> g:=SymmetricGroup(4);; gap> PCore(g,2); Group([ (1,4)(2,3), (1,2)(3,4) ]) gap> PCore(g,2) = Core(g,SylowSubgroup(g,2)); true

`‣ NormalClosure` ( G, U ) | ( operation ) |

`‣ NormalClosure` ( G, list ) | ( operation ) |

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

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

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

`‣ 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) ])

`‣ 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) ]) ]

`‣ InfoComplement` | ( info class ) |

Info class for the complement routines.

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).

`‣ TrivialSubgroup` ( G ) | ( attribute ) |

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

`‣ 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 \(g \in \textit{G}\) and \(h \in \textit{H}\), that is the group \(\left \langle [ g, h ] \mid g \in \textit{G}, h \in \textit{H} \right \rangle\).

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

`‣ DerivedSubgroup` ( G ) | ( attribute ) |

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

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

`‣ 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

`‣ 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) ])

`‣ FrattiniSubgroup` ( G ) | ( attribute ) |

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

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

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

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

`‣ PerfectResiduum` ( G ) | ( attribute ) |

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

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

`‣ SolvableRadical` ( G ) | ( attribute ) |

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

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

`‣ 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) ])

`‣ SupersolvableResiduum` ( G ) | ( attribute ) |

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

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

`‣ PRump` ( G, p ) | ( operation ) |

For a prime \(p\), the * p-rump* of a group

`PCentralSeries`

(39.17-13).gap> g:=DicyclicGroup(12);; gap> PRump(g,2) = PCentralSeries(g,2)[2]; true gap> g:=SymmetricGroup(4);; gap> PRump(g,2) = AlternatingGroup(4); true

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.

For each prime \(p\) dividing the order of \(G\), there exists a \(p\)-complement (see

`SylowComplement`

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

`HallSubgroup`

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

`SylowSystem`

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

`ComplementSystem`

(39.13-5)).

`‣ SylowSubgroup` ( G, p ) | ( operation ) |

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) ])

`‣ SylowComplement` ( G, p ) | ( operation ) |

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

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

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

`‣ HallSubgroup` ( G, P ) | ( operation ) |

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

.

If `G` is solvable this function always returns a subgroup. If `G` is not solvable this function might return a subgroup (if it is unique up to conjugacy), a list of subgroups (which are representatives of the conjugacy classes in case there are several such classes) or `fail`

if no such subgroup exists.

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

`‣ 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 ]

`‣ 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 \(|\textit{G}|\) that omit exactly one prime. Every pair of subgroups from this set commutes as subgroups. Complement systems exist only for solvable groups, therefore `ComplementSystem`

returns `fail`

if the group `G` is not solvable.

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

`‣ 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 ]

`‣ Omega` ( G, p[, n] ) | ( operation ) |

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

.

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

`‣ Agemo` ( G, p[, n] ) | ( function ) |

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

.

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

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`

(39.15-25) and `IsPNilpotent`

(39.15-26), 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.

`‣ 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).

`‣ 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 \(p\).

`‣ 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.

`‣ 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.

`‣ IsPerfectGroup` ( G ) | ( property ) |

A group is *perfect* if it equals its derived subgroup (see `DerivedSubgroup`

(39.12-3)).

`‣ 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.

`‣ 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)).

`‣ IsSupersolvableGroup` ( G ) | ( property ) |

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

`‣ IsMonomialGroup` ( G ) | ( property ) |

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

`‣ IsSimpleGroup` ( G ) | ( property ) |

`‣ IsNonabelianSimpleGroup` ( G ) | ( property ) |

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

`‣ 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

`‣ IsQuasisimpleGroup` ( G ) | ( property ) |

A group `G` is *quasisimple* if `G` is perfect (see `IsPerfectGroup`

(39.15-5)) and if `G`\(/Z(\)`G`\()\) is simple (see `IsSimpleGroup`

(39.15-10)), where \(Z(\)`G`\()\) is the centre of `G` (see `Centre`

(35.4-5)).

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

`‣ IsomorphismTypeInfoFiniteSimpleGroup` ( G ) | ( attribute ) |

`‣ IsomorphismTypeInfoFiniteSimpleGroup` ( n ) | ( attribute ) |

For a finite simple group `G`, `IsomorphismTypeInfoFiniteSimpleGroup`

returns a record with the components `name`

, `shortname`

, `series`

, and possibly `parameter`

, describing the isomorphism type of `G`.

The values of the components `name`

, `shortname`

, and `series`

are strings, `name`

gives name(s) for `G`, `shortname`

gives one name for `G` that is compatible with the naming scheme used in the **GAP** packages **CTblLib** and **AtlasRep** (and in the **Atlas** of Finite Groups [CCN+85]), and `series`

describes the following series.

(If different characterizations of `G` are possible only one is given by `series`

and `parameter`

, while `name`

may give several names.)

`"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", shortname := "L3(2)" )

For a positive integer `n`, `IsomorphismTypeInfoFiniteSimpleGroup`

returns `fail`

if `n` is not the order of a finite simple group, and a record as described for the case of a group `G` otherwise. If more than one simple group of order `n` exists then the result record contains only the `name`

component, a string that lists the two possible isomorphism types of simple groups of this order.

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

`‣ SimpleGroup` ( id[, param] ) | ( function ) |

This function will construct **an** instance of the specified nonabelian simple group. Groups are specified via their name in ATLAS style notation, with parameters added if necessary. The intelligence applied to parsing the name is limited, and at the moment no proper extensions can be constructed. For groups 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)

`‣ SimpleGroupsIterator` ( [start[, end]] ) | ( function ) |

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

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

`‣ SmallSimpleGroup` ( order[, i] ) | ( function ) |

Returns: The `i`th 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)

`‣ 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)" ]

`‣ 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.

`‣ 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.

`‣ IsPGroup` ( G ) | ( property ) |

A *\(p\)-group* is a group in which the order (see `Order`

(31.10-10)) of every element is of the form \(p^n\) for a prime integer \(p\) and a nonnegative integer \(n\). `IsPGroup`

returns `true`

if `G` is a \(p\)-group, and `false`

otherwise.

Finite \(p\)-groups are precisely those groups whose order (see `Size`

(30.4-6)) is a prime power, and are always nilpotent.

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

`‣ IsPowerfulPGroup` ( G ) | ( property ) |

A finite p-group `G` is said to be a *powerful \(p\)-group* if the commutator subgroup \([\textit{G},\textit{G}]\) is contained in \(\textit{G}^{p}\) if the prime \(p\) is odd, or if \([\textit{G},\textit{G}]\) is contained in \(\textit{G}^{4}\) if \(p = 2\). The subgroup \(\textit{G}^{p}\) is called the first Agemo subgroup, (see `Agemo`

(39.14-2)). `IsPowerfulPGroup`

returns `true`

if `G` is a powerful \(p\)-group, and `false`

otherwise. *Note: *This function returns `true`

if `G` is the trivial group.

`‣ PrimePGroup` ( G ) | ( attribute ) |

If `G` is a nontrivial \(p\)-group (see `IsPGroup`

(39.15-20)), `PrimePGroup`

returns the prime integer \(p\); if `G` is trivial then `PrimePGroup`

returns `fail`

. Otherwise an error is issued.

(One should avoid a common error of writing `if IsPGroup(g) then ... PrimePGroup(g) ...`

where the code represented by dots assumes that `PrimePGroup(g)`

is an integer.)

`‣ PClassPGroup` ( G ) | ( attribute ) |

The \(p\)-class of a \(p\)-group `G` (see `IsPGroup`

(39.15-20)) is the length of the lower \(p\)-central series (see `PCentralSeries`

(39.17-13)) of `G`. If `G` is not a \(p\)-group then an error is issued.

`‣ RankPGroup` ( G ) | ( attribute ) |

For a \(p\)-group `G` (see `IsPGroup`

(39.15-20)), `RankPGroup`

returns the *rank* of `G`, which is defined as the minimal size of a generating system of `G`. If `G` is not a \(p\)-group then an error is issued.

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

`‣ IsPSolvable` ( G, p ) | ( operation ) |

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

`‣ IsPNilpotent` ( G, p ) | ( operation ) |

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

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.

`‣ 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 \(\textit{G}/\textit{G}'\) as a direct product of cyclic groups of prime power (or infinite) order.

(See `IndependentGeneratorsOfAbelianGroup`

(39.22-5) to obtain actual generators).

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

`‣ 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 \(g \in \textit{G}\).

gap> Exponent(g); 12

`‣ EulerianFunction` ( G, n ) | ( operation ) |

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

In [Hal36], the notation \(\phi_{\textit{n}}(\textit{G})\) is used for the value returned by `EulerianFunction`

, and the quotient of \(\phi_{\textit{n}}(\textit{G})\) by the order of the automorphism group of `G` is called \(d_{\textit{n}}(\textit{G})\). If `G` is a nonabelian simple group then \(d_{\textit{n}}(\textit{G})\) is the greatest number \(d\) for which the direct product of \(d\) groups isomorphic with `G` can be generated by `n` elements.

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

(70.9-9).

gap> EulerianFunction( g, 2 ); 432

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

A series 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.

`‣ 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(()) ]

`‣ 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).

`‣ 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).

`‣ 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 \(V \geq \)`U`. If `U` is subnormal, \(V =\) `U`.

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

`‣ CompositionSeries` ( G ) | ( attribute ) |

`‣ CompositionSeriesThrough` ( G, normals ) | ( operation ) |

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

takes as second argument a list `normals` of normal subgroups of the group, and returns a composition series that incorporates these normal subgroups.

`‣ 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) | C2 S (5 gens, size 2520) | A7 1 (0 gens, size 1)

`‣ DerivedSeriesOfGroup` ( G ) | ( attribute ) |

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

`‣ 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

`‣ 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 ]

`‣ InvariantElementaryAbelianSeries` ( G, morph[, N[, fine]] ) | ( function ) |

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(()) ]

`‣ LowerCentralSeriesOfGroup` ( G ) | ( attribute ) |

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

`‣ UpperCentralSeriesOfGroup` ( G ) | ( attribute ) |

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

`‣ PCentralSeries` ( G, p ) | ( operation ) |

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

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

`‣ JenningsSeries` ( G ) | ( attribute ) |

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

`‣ 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 ]

`‣ 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.

`‣ IntermediateGroup` ( G, U ) | ( function ) |

This routine tries to find a subgroup \(E\) of `G`, such that \(\textit{G} > E > \textit{U}\) holds. If `U` is maximal in `G`, the function returns `fail`

. This is done by finding minimal blocks for the operation of `G` on the right cosets of `U`.

`‣ 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.

`‣ StructuralSeriesOfGroup` ( G ) | ( attribute ) |

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

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

`‣ 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 \(\textit{G}/\textit{N}\) can be identified with the image of `G` this permits efficient computations in the factor group. The homomorphism returned is not necessarily surjective, so `ImagesSource`

(32.4-1) should be used instead of `Range`

(32.3-7) to get a group isomorphic to the factor group. The `NC`

variant does not check whether `N` is normal in `G`.

`‣ FactorGroup` ( G, N ) | ( function ) |

`‣ FactorGroupNC` ( G, N ) | ( operation ) |

returns the image of the `NaturalHomomorphismByNormalSubgroup(`

. This function is provided for compatibility with older code, but if a connection between group and factor is desired, users need to start by obtaining the `G`,`N`)`NaturalHomomorphismByNormalSubgroup`

in the first place. The `NC`

version does not test whether `N` is normal in `G`.

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

`‣ CommutatorFactorGroup` ( G ) | ( attribute ) |

computes the commutator factor group \(\textit{G}/\textit{G}'\) of the group `G`.

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

`‣ 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).

`‣ HasAbelianFactorGroup` ( G, N ) | ( function ) |

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

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

`‣ HasElementaryAbelianFactorGroup` ( G, N ) | ( function ) |

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

`‣ CentralizerModulo` ( G, N, elm ) | ( operation ) |

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

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

`‣ 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) ])

`‣ 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.

`‣ 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 ]

`‣ ConjugacyClassesMaximalSubgroups` ( G ) | ( attribute ) |

returns the conjugacy classes of maximal subgroups of `G`. Representatives of the classes can be computed directly by `MaximalSubgroupClassReps`

(39.19-5).

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

`‣ MaximalSubgroupClassReps` ( G ) | ( attribute ) |

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

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

`‣ LowIndexSubgroups` ( G, index ) | ( operation ) |

The operation `LowIndexSubgroups`

computes representatives of the conjugacy classes of subgroups of the group `G` that index less than or equal to `index`.

For finitely presented groups this operation simply defaults to `LowIndexSubgroupsFpGroup`

(47.10-1). In other cases, it uses repeated calculation of maximal subgroups.

The function `LowLayerSubgroups`

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

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

`‣ 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) ]) ]

`‣ 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) ]) ]

`‣ NormalSubgroups` ( G ) | ( attribute ) |

returns a list of all normal subgroups of `G`.

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

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

`‣ MaximalNormalSubgroups` ( G ) | ( attribute ) |

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

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

`‣ MinimalNormalSubgroups` ( G ) | ( attribute ) |

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

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

`‣ CharacteristicSubgroups` ( G ) | ( attribute ) |

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

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

`‣ 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 ]

`‣ 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.

`‣ 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 https://www.graphviz.org. (On the Macintosh, the program `OmniGraffle`

is also able to read this format.)

Subgroups are labelled in the form

where `i`-`j``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");

`‣ 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

`‣ 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

`‣ LowLayerSubgroups` ( [act, ]G, lim[, cond, dosub] ) | ( function ) |

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

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

`‣ ContainedConjugates` ( G, A, B[, onlyone] ) | ( operation ) |

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

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

`‣ ContainingConjugates` ( G, A, B ) | ( operation ) |

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

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

`‣ MinimalFaithfulPermutationDegree` ( G ) | ( operation ) |

`‣ MinimalFaithfulPermutationRepresentation` ( G ) | ( operation ) |

For a finite group `G`, `MinimalFaithfulPermutationDegree`

calculates the least positive integer \(n=\mu(G)\) such that `G` is isomorphic to a subgroup of the symmetric group of degree \(n\). This can require calculating the whole subgroup lattice. The operation `MinimalFaithfulPermutationRepresentation`

returns a corresponding isomorphism.

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

`‣ 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`

(50.6-2)), thus it will issue an error if the library is insufficient to determine all perfect subgroups.

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

`‣ ConjugacyClassesPerfectSubgroups` ( G ) | ( attribute ) |

returns a list of the conjugacy classes of perfect subgroups of `G`. (see `RepresentativesPerfectSubgroups`

(39.20-10).)

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

`‣ 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.

`‣ InfoLattice` | ( info class ) |

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

`‣ 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>

##

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

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

`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 \(F \geq C > A > N > B > M\). \(N\) 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 \(U \cap N = B\) and \(\langle U, N \rangle = A\), 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 \(F = \textit{G}/H\). 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> SortedList(List(l,Size)); [ 1, 2, 3, 4, 6, 8, 9, 18, 27, 54, 108, 216 ] gap> Length(ConjugacyClassesSubgroups(g)); # to compare 162

`‣ SizeConsiderFunction` ( size ) | ( function ) |

This function returns a function `consider`

of five arguments that can be used in `SubgroupsSolvableGroup`

(39.21-3) for the option `consider`

to compute subgroups whose sizes are divisible by `size`.

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

This example shows that in general the `consider`

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

`‣ ExactSizeConsiderFunction` ( size ) | ( function ) |

This function returns a function `consider`

of five arguments that can be used in `SubgroupsSolvableGroup`

(39.21-3) for the option `consider`

to compute subgroups whose sizes are exactly `size`.

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

Again, the `consider`

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

`‣ InfoPcSubgroup` | ( info class ) |

Information function for the subgroup lattice functions using pcgs.

`‣ GeneratorsSmallest` ( G ) | ( attribute ) |

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

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

`‣ LargestElementGroup` ( G ) | ( attribute ) |

returns the largest element of `G` with respect to the ordering `<`

of the elements family.

`‣ 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) ]

`‣ 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) ]

`‣ IndependentGeneratorsOfAbelianGroup` ( A ) | ( attribute ) |

returns a list of generators \(a_1, a_2, \ldots\) of prime power order or infinite order of the abelian group `A` such that `A` is the direct product of the cyclic groups generated by the \(a_i\). The list of orders of the returned generators must match the result of `AbelianInvariants`

(39.16-1) (taking into account that zero and `infinity`

(18.2-1) are identified).

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

`‣ IndependentGeneratorExponents` ( G, g ) | ( operation ) |

For an abelian group `G`, with `IndependentGeneratorsOfAbelianGroup`

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

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

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

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

and is a \(GF(p)\)-vector space.

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

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

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

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

`‣ OneCocycles` ( G, M ) | ( function ) |

`‣ OneCocycles` ( G, mpcgs ) | ( function ) |

`‣ OneCocycles` ( gens, M ) | ( function ) |

`‣ OneCocycles` ( gens, mpcgs ) | ( function ) |

Computes the group of 1-cocycles \(Z^1(\textit{G}/\textit{M},\textit{M})\). The normal subgroup `M` may be given by a (Modulo)Pcgs `mpcgs`. In this case the whole calculation is performed modulo the normal subgroup defined by `DenominatorOfModuloPcgs(`

(see 45.1). Similarly the group `mpcgs`)`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.

`‣ OneCoboundaries` ( G, M ) | ( function ) |

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

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

(39.23-3).

`‣ OCOneCocycles` ( ocr, onlySplit ) | ( function ) |

is the more technical function to compute 1-cocycles. It takes a record `ocr` as first argument which must contain at least the components `group`

for the group and `modulePcgs`

for a (modulo) pcgs of the module. This record will also be returned with components as described under `OneCocycles`

(39.23-1) (with the exception of `isSplitExtension`

which is indicated by the existence of a `complement`

) but components such as `oneCoboundaries`

will only be computed if not already present.

If `onlySplit` is `true`

, `OCOneCocycles`

returns `false`

as soon as possible if the extension does not split.

`‣ ComplementClassesRepresentativesEA` ( G, N ) | ( function ) |

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`.

`‣ InfoCoh` | ( info class ) |

The info class for the cohomology calculations is `InfoCoh`

.

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].

`‣ EpimorphismSchurCover` ( G[, pl] ) | ( attribute ) |

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.

`‣ SchurCover` ( G ) | ( attribute ) |

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

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.

`‣ 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.

`‣ 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 \(\textit{G}/N\) is a central quotient of a group. It is also equal to the Exterior Center of `G`, see [Ell98].

`‣ NonabelianExteriorSquare` ( G ) | ( operation ) |

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

`‣ EpimorphismNonabelianExteriorSquare` ( G ) | ( operation ) |

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

`‣ 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].

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) > ]

`‣ 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 \(\textit{p} \geq 0\). There are two such representations and `sign` controls which is returned: +1 gives a group where the preimage of an adjacent transposition \((i,i+1)\) has order 4, -1 gives a group where the preimage of an adjacent transposition \((i,i+1)\) has order 2. If no `sign` is specified, +1 is used by default. If no `p` is specified, 3 is used by default. (Note that the convention of which cover is labelled as +1 is inconsistent in the literature.)

`‣ SchurCoverOfSymmetricGroup` ( n, p, sign ) | ( operation ) |

Constructs a Schur cover of `SymmetricGroup(`

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

is returned if \(\textit{p} = 2\). For \(\textit{p} = 3\), \(\textit{n} = 3\), the representation is indecomposable, but reducible. The field of the matrix group is generally `GF(`

if \(\textit{p} > 0\), and an abelian number field if \(\textit{p} = 0\).`p`^2)

`‣ DoubleCoverOfAlternatingGroup` ( n, p ) | ( operation ) |

Constructs a double cover of `AlternatingGroup(`

as a faithful, completely reducible matrix group in characteristic `n`)`p` (\(p \neq 2\)) for \(n \geq 4\). For \(n \leq 3\), the alternating group is its own Schur cover, and `fail`

is returned. For \(p = 2\), there is no faithful, completely reducible representation of the double cover, so `fail`

is returned. The field of the matrix group is generally `GF(p^2)`

if \(p>0\), and an abelian number field if \(p=0\). If `p` is omitted, the default is 3.

`‣ TwoCohomologyGeneric` ( G, M ) | ( operation ) |

This function computes the second cohomology group for an arbitrary finite group `G`. The generators of the module `M` must correspond to the generators of `G`. It returns a record with components `coboundaries`

, `cocycles`

and `cohomology`

which are lists of vectors that form a basis of the respective group. `cohomology`

is chosen as a vector space complement to `coboundaries`

in the `cocycles`

. These vectors are representing tails in `M` with respect to the `relators`

of the presentation `presentation`

of `G`. (Note that this presentation is on a generating set chosen by the routine, this generating system corresponds to the components `group`

and `module`

of the record returned. The extension corresponding to a cocyle `c`

can be constructed as `FpGroupCocycle(r,c)`

where `r`

is the cohomology record. This is currently done as a finitely presented group.

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

`‣ FpGroupCocycle` ( r, c[, doperm] ) | ( function ) |

For a record `r` as returned by `TwoCohomologyGeneric`

(39.25-1) and a vector `c` in the space of two-cocycles, this operation returns a finitely presented group that is an extension corresponding to the cocycle `c`. If the underlying module has dimension \(d\), the last \(d\) generators generate the normal subgroup. If the optional parameter `doperm` is given as `true`, a faithful permutation representation is computed and stored in the attribute `IsomorphismPermGroup`

(43.3-1) of the computed group. If the option `normalform`

is given as `true`

, arithmetic in the resulting finitely presented group will bring words into normal form.

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

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

Also see Section 46.8 for operations and methods specific for Pc groups.

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).

`‣ CanEasilyTestMembership` ( G ) | ( filter ) |

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.

`‣ CanEasilyComputeWithIndependentGensAbelianGroup` ( G ) | ( filter ) |

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.

`‣ CanComputeSize` ( dom ) | ( filter ) |

This filter indicates that we know that the size of the domain `dom` (which might be `infinity`

(18.2-1)) can be computed reasonably easily. It doesn't imply as quick a computation as `HasSize`

would but its absence does not imply that the size cannot be computed.

`‣ CanComputeSizeAnySubgroup` ( G ) | ( filter ) |

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.)

`‣ CanComputeIndex` ( G, H ) | ( operation ) |

This function indicates whether the index \([\textit{G}:\textit{H}]\) (which might be `infinity`

(18.2-1)) can be computed. It assumes that \(\textit{H} \leq \textit{G}\) (see `CanComputeIsSubset`

(39.26-6)).

`‣ CanComputeIsSubset` ( A, B ) | ( operation ) |

This filter indicates that **GAP** can test (via `IsSubset`

(30.5-1)) whether `B` is a subset of `A`.

`‣ 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.

`‣ NormalizerViaRadical` ( G, S ) | ( function ) |

This function implements a particular approach, following the SolvableRadical paradigm, for calculating the normalizer of a subgroup `S` in `G`. It is at the moment provided only as a separate function, and not as method for the operation `Normalizer`

, as it can often be slower than other built-in routines. In certain hard cases (non-solvable groups with nontrivial radical), however its performance is substantially superior. The function thus is provided as a non-automated tool for advanced users.

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

Note that this example only demonstrates usage, but that in this case in fact the ordinary `Normalizer`

routine performs faster.

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