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# 37 Groups

### Sections

1. Group Elements
2. Creating Groups
3. Subgroups
4. Closures of (Sub)groups
5. Expressing Group Elements as Words in Generators
6. Structure Descriptions
7. Cosets
8. Transversals
9. Double Cosets
10. Conjugacy Classes
11. Normal Structure
12. Specific and Parametrized Subgroups
13. Sylow Subgroups and Hall Subgroups
14. Subgroups characterized by prime powers
15. Group Properties
16. Numerical Group Attributes
17. Subgroup Series
18. Factor Groups
19. Sets of Subgroups
20. Subgroup Lattice
21. Specific Methods for Subgroup Lattice Computations
22. Special Generating Sets
23. 1-Cohomology
24. Schur Covers and Multipliers
25. Tests for the Availability of Methods

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 S4 acting on the letters {1,…,4}.

## 37.1 Group Elements

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

    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` (see Size). For convenience, group orders can also be computed with `Order`.

The operation `Comm` (see Comm) can be used to compute the commutator of two elements, the operation `LeftQuotient` (see LeftQuotient) computes the product x−1y.

## 37.2 Creating Groups

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

- `Group(` gen`, ... ) F`
- `Group(` gens`) F`
- `Group(` gens`,` id`) F`

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

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

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

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

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

- `GroupWithGenerators(` gens`) O`
- `GroupWithGenerators(` gens`,` id`) O`

  `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` of G is equal to gens.

- `GeneratorsOfGroup(` G`) A`

  returns a list of generators of the group G. If G has been created by the command `GroupWithGenerators` (see GroupWithGenerators), with argument gens, then the list returned by `GeneratorsOfGroup` will be equal to gens.

      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` (see View) will in general display only some information about the group which may even be just the fact that it is a group.

- `AsGroup(` D`) A`

  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`) O`

  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 use the infix operator `^`.

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

- `IsGroup(` obj`) C`

  A group is a magma-with-inverses (see IsMagmaWithInverses) and associative (see IsAssociative) 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` (see AsGroup) if you want to perform such a test. (See Categories for details about categories.)

      gap> IsGroup(g);
      true

- `InfoGroup V`

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

  ## 37.3 Subgroups

  For the general concept of parents and subdomains, see Parents and Constructing Subdomains. More functions that construct certain subgroups can be found in the sections Normal Structure, Specific and Parametrized Subgroups, Sylow Subgroups and Hall Subgroups, and Subgroups characterized by prime powers.

- `Subgroup(` G`,` gens`) F`
- `SubgroupNC(` G`,` gens`) F`

  creates the subgroup U of G generated by gens. The `Parent` of U will be G. The `NC` version does not check, whether the elements in gens actually lie in G.

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

- `Index(` G`,` U`) O`
- `IndexNC(` G`,` U`) O`

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

      gap> Index(g,u);
      4

- `IndexInWholeGroup(` G`) A`

  If the family of elements of G itself forms a group P, this attribute returns the index of G in P.

- `AsSubgroup(` G`,` U`) O`

  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`) F`

  `IsSubgroup` returns `true` if U is a group that is a subset of the domain G. This is actually checked by calling `IsGroup(` U`)` and `IsSubset(` G`,` U`)`; note that special methods for `IsSubset` (see IsSubset) 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`.

      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`) O`

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

  A group G normalizes a group U if and only if for every g ∈ G and u ∈ U the element ug is a member of U. Note that U need not be a subgroup of G.

      gap> IsNormal(g,u);
      false

- `IsCharacteristicSubgroup(` G`,` N`) O`

  tests whether N is invariant under all automorphisms of G.

      gap> IsCharacteristicSubgroup(g,u);
      false

- `ConjugateSubgroup(` G`,` g`) O`

- `ConjugateSubgroups(` G`,` U`) O`

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

- `IsSubnormal(` G`,` U`) O`

  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

  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`) with the parent as an attribute.

- `SubgroupByProperty(` G`,` prop`) F`

  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`) F`

  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>

  ## 37.4 Closures of (Sub)groups

- `ClosureGroup(` G`,` obj`) O`

  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`) F`
- `ClosureGroupCompare(` G`,` elm`) F`
- `ClosureGroupIntest(` G`,` elm`) F`

  These three functions together with `ClosureGroupDefault` implement the main methods for `ClosureGroup` (see ClosureGroup). 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`) F`

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

- `ClosureSubgroup(` G`,` obj`) F`
- `ClosureSubgroupNC(` G`,` obj`) F`

  For a group G that stores a parent group (see Parents), `ClosureSubgroup` calls `ClosureGroup` (see ClosureGroup) 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.

  ## 37.5 Expressing Group Elements as Words in Generators

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

  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 S4 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^-2*y^-1*x^-1*y^-1*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 S16.

      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`) F`

  returns a factorization of elm as word in the generators of G given in the attribute `GeneratorsOfGroup`. The component ``G`!.factFreeMap` will contain a map map from the group G to the free group in which the word is expressed. The attribute `MappingGeneratorsImages` of this map gives a list of generators and corresponding letters.

  The algorithm used computes 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 thousand 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 a 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*x2*x1*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) ] ]

  ## 37.6 Structure Descriptions

- `StructureDescription(` G`) A`

  The method for `StructureDescription` exhibits the structure of the given group to some extend 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 and can be computed reasonably quickly.

  Note that non-isomorphic groups can have the same `StructureDescription`, since the structure description might not exhibit the structure of the considered group in all detail. However, isomorphic groups in different representation will always obtain the same structure description.

  The `StructureDescription` is a string of the following form:

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

  Note that the method chooses one possible way of building up the given group from smaller pieces (others are possible too).

  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` attribute is not bound, but `StructureDescription` is, `View` prints the value of the attribute `StructureDescription`. The `Print`ed representation of a group is not affected by computing a `StructureDescription`.

  The strategy is

  1.
    Lookup in precomputed list, if the order of G is not larger than 100 and not equal to 64.

  2.
    If G is abelian: decompose it into cyclic factors in ``elementary divisors style'', e.g. `"C2 x C3 x C3"` is `"C6 x C3"`.

  3.
    Recognize alternating groups, symmetric groups, dihedral groups, quasidihedral groups, quaternion groups, PSL's, SL's, GL's and simple groups not listed so far as basic building blocks.

  4.
    Decompose into a direct product of irreducible factors.

  5.
    Recognize semidirect products (N:H), where N is normal. Select a pair N, H with the following preferences:
    1.
      H is abelian

    2.
      N is abelian

    2a.
      N has many abelian invariants

    3.
      N is a direct product

    3a.
      N has many direct factors

    4.
      φ: H → Aut(N), h → (n → nh) is injective.

  6.
    Fall back to non-splitting extensions: If the centre or the commutator factor group is non-trivial, write G as Z(G).G/Z(G) resp. G'.G/G'. Otherwise if the Frattini subgroup is non-trivial, write G as Φ(G).G/Φ(G).

  7.
    If no decomposition is found (maybe this is not the case for any finite group) try to identify G in the perfect groups library. If also this fails return a string describing this situation.

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

    ## 37.7 Cosets

- `RightCoset(` U`,` g`) O`

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

  Right cosets are external orbits for the action of U which acts via `OnLeftInverse`. Of course the action of a larger group G on right cosets is via `OnRight`.

      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)(2,4), (2,4), (1,4,2,3), (1,3,4,2) ]

- `RightCosets(` G`,` U`) F`
- `RightCosetsNC(` G`,` U`) O`

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

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

- `CanonicalRightCosetElement( U, g ) O`

  returns a ``canonical'' representative of the coset Ug 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`) C`

  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.

  ## 37.8 Transversals

- `RightTransversal(` G`,` U`) O`

  A right transversal t is a list of representatives for the set U \ 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 [G :U ], except if U is the trivial subgroup of G in which case `RightTransversal` may return the sorted plain list of coset representatives.

  The operation `PositionCanonical(`t`,`g`)`, 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` will return `fail` if the element is not equal to the representative.) Functions that implement group actions such as `Action` or `Permutation` (see Chapter Group Actions) use `PositionCanonical`, 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` (see CanonicalRightCosetElement), but we may compute a list of canonical coset representatives by calling that function.

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

  The operation `PositionCanonical` is described in section PositionCanonical.

      gap> PositionCanonical(rt,(1,2,3));
      1
      gap> rt[1];
      ()

  ## 37.9 Double Cosets

- `DoubleCoset(` U`,` g`,` V`) O`

  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 UgV which is the set of all elements of the form ugv for any u ∈ U , v ∈ V . For element operations such as `in`, a double coset behaves like a set of group elements. The double coset stores U in the attribute `LeftActingGroup`, g as `Representative`, and V as `RightActingGroup`.

- `RepresentativesContainedRightCosets(` D`) A`

  A double coset UgV can be considered as an union of right cosets U hi. (it is the union of the orbit of Ug under right multiplication by V.) For a double coset D=UgV this returns a set of representatives hi such that D =∪hiU hi. The representatives returned are canonical for U (see CanonicalRightCosetElement) 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`) O`
- `DoubleCosetsNC(` G`,` U`,` V`) O`

  computes a duplicate free list of all double cosets UgV for g ∈ G . U and V must be subgroups of the group G. The NC version does not check whether U and V are both 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`) C`

  The category of double cosets.

- `DoubleCosetRepsAndSizes(` G`,` U`,` V`) O`

  returns a list of double coset representatives and their sizes, the entries are lists of the form [rep ,size ]. This operation is faster that `DoubleCosetsNC` 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 V`

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

  ## 37.10 Conjugacy Classes

- `ConjugacyClass(` G`,` g`) O`

  creates the conjugacy class in G with representative g. This class is an external set, so functions such as `Representative` (which returns g), `ActingDomain` (which returns G), `StabilizerOfExternalSet` (which returns the centralizer of g) and `AsList` work for it.

  A conjugacy class is an external orbit (ExternalOrbit) of group elements with the group acting by conjugation on it. Thus element tests or operation representatives can be computed. The attribute `Centralizer` gives the centralizer of the representative (which is the same result as `StabilizerOfExternalSet`). (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`.)

- `ConjugacyClasses(` G`) A`

  returns the conjugacy classes of elements of G as a list of `ConjugacyClass`es of G (see `ConjugacyClass` (ConjugacyClass) 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). 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 Conjugacy Classes in Solvable Groups).

  For other groups the method of HulpkeClasses 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` (ConjugacyClassesByRandomSearch) below.

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

  `noaction`
    Even for small groups `ConjugacyClassesByOrbits` (ConjugacyClassesByOrbits) 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`) F`

  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 `ConjugacyClass`.

- `ConjugacyClassesByOrbits(` G`) F`

  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 `ConjugacyClass`.

  Typically, for small groups (roughly of order up to 103) 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 folowing 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`) A`

  returns the number of conjugacy classes of G.

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

- `RationalClass(` G`,` g`) O`

  creates the rational class in G with representative g. A rational class consists of all elements that are conjugate to g or to a power gi 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) of group elements with the group acting by conjugation on it, but not an external orbit.

- `RationalClasses(` G`) A`

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

      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`) A`

  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 Pn (see PrimitiveRootMod), the so-called Galois group of the rational class. The prime residue class group Pn is obtained in GAP as `Units( Integers mod` n`)`, the unit group of a residue class ring. The Galois group of a rational class rcl is stored in the attribute `GaloisGroup(`rcl`)` as a subgroup of this group.

- `IsConjugate(` G`,` x`,` y`) O`
- `IsConjugate(` G`,` U`,` V`) O`

  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`.

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

  `RepresentativeAction` (see RepresentativeAction) can be used to obtain conjugating elements.

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

  ## 37.11 Normal Structure

  For the operations `Centralizer` and `Centre`, see Chapter Magmas.

- `Normalizer(` G`,` U`) O`
- `Normalizer(` G`,` g`) O`

  Computes the normalizer NG(U), that is the stabilizer of U under the conjugation action of G. The second form computes NG(〈g〉).

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

- `Core(` S`,` U`) O`

  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`) F`

  The p-core of G is the largest normal p-subgroup of G. It is the core of a p-Sylow subgroup of G.

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

- `NormalClosure(` G`,` U`) O`

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

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

- `NormalIntersection(` G`,` U`) O`

  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) ])

- `Complementclasses(` G`,` N`) O`

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

- `InfoComplement V`

  Info class for the complement routines.

  ## 37.12 Specific and Parametrized Subgroups

  The Centre of a group (the subgroup of those elements that commute with all other elements of the group) can be computed by the operation `Centre` (see Centre).

- `TrivialSubgroup(` G`) A`

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

- `CommutatorSubgroup(` G`,` H`) O`

  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) of elements g ∈ G and h ∈ H , that is the group 〈 [ g, h ] | g ∈ G , h ∈ H 〉.

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

- `DerivedSubgroup(` G`) A`

  The derived subgroup G′ of G is the subgroup generated by all commutators of pairs of elements of G. It is normal in G and the factor group G/G′ is the largest abelian factor group of G.

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

- `CommutatorLength(` G`) A`

  returns the minimal number n such that each element in the derived subgroup (see DerivedSubgroup) 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`) A`

  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`) A`

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

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

- `PrefrattiniSubgroup(` G`) A`

  returns a Prefrattini subgroup of the finite solvable group G. A factor M/N of G is called a Frattini factor if M/N ≤ φ(G/N) holds. The group 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`) A`

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

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

- `RadicalGroup(` G`) A`

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

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

- `Socle(` G`) A`

  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`) A`

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

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

- `PRump(` G`,` p`) F`

  The p-rump of a group G is the subgroup G′Gp for a prime p.

  @example missing!@

  ## 37.13 Sylow Subgroups and Hall Subgroups

- `SylowSubgroup(` G`,` p`) F`

  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) ])

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

  1. For each prime p dividing the order of G, there exists a p-complement (see SylowComplement).
  2. For each set P of primes dividing the order of G, there exists a P-Hall subgroup (see HallSubgroup).
  3. G has a Sylow system (see SylowSystem).
  4. G has a complement system (see ComplementSystem).

- `SylowComplement(` G`,` p`) F`

  returns a p-Sylow complement of the finite group G. This is a subgroup U of order coprime to p such that the index [G:U] is a p-power. At the moment methods exist only if G is solvable and GAP will issue an error if G is not solvable.

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

- `HallSubgroup(` G`,` P`) F`

  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. The function computes Hall subgroups via the operation `HallSubgroupOp`. At the moment methods exist only if G is solvable and GAP will issue an error if G is not solvable.

      gap> h:=SmallGroup(60,10);;
      gap> u:=HallSubgroup(h,[2,3]);
      Group([ f1, f2, f3 ])
      gap> Size(u);
      12

- `SylowSystem(` G`) A`

  A Sylow system of a group G is a set of Sylow subgroups of G such that every pair of Sylow 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`) A`

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

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

- `HallSystem(` G`) A`

  returns a list containing one Hall-P subgroup for each set P of primes which occur in 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 ]

  ## 37.14 Subgroups characterized by prime powers

- `Omega(` G`,` p`[,` n`] ) F`

  For a p-group G, one defines Ωn (G ) = { g ∈ G | gp n = 1 }. The default value for n is `1`.

  @At the moment methods exist only for abelian G and n=1.@

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

- `Agemo(` G`,` p`[,` n`] ) F`

  For a p-group G, one defines \mhon (G) = 〈gp n | g ∈ G 〉. The default value for n is `1`.

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

  ## 37.15 Group Properties

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

- `IsCyclic(` G`) P`

  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` (see MinimalGeneratingSet).

- `IsElementaryAbelian(` G`) P`

  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`) P`

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

- `NilpotencyClassOfGroup(` G`) A`

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

  If G is not nilpotent an error is issued.

- `IsPerfectGroup(` G`) P`

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

- `IsSolvableGroup(` G`) P`

  A group is solvable if the derived series (see DerivedSeriesOfGroup 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), and each polycyclic group is solvable, but there are infinite solvable groups that are not polycyclic.

- `IsPolycyclicGroup(` G`) P`

  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).

- `IsSupersolvableGroup(` G`) P`

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

- `IsMonomialGroup(` G`) P`

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

- `IsSimpleGroup(` G`) P`

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

- `IsomorphismTypeInfoFiniteSimpleGroup(` G`) F`

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

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

  `"A"`
    Alternating groups, `parameter` gives the natural degree.

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

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

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

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

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

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

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

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

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

  `"2E"`
    Twisted exceptional Chevalley type E6, `parameter` is a value q that indicates 2E6(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 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( series := "L", parameter := [ 2, 7 ],
        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)" )

- `IsFinitelyGeneratedGroup(` G`) P`

  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`) P`

  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`) P`

  A p-group is a finite group whose order (see Size) is of the form pn for a prime integer p and a nonnegative integer n. `IsPGroup` returns `true` if G is a p-group, and `false` otherwise.

- `PrimePGroup(` G`) A`

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

- `PClassPGroup(` G`) A`

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

- `RankPGroup(` G`) A`

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

  Note that the following functions, although they are mathematical properties, are not properties in the sense of GAP (see Attributes and Properties), as they depend on a parameter.

- `IsPSolvable(` G`,` p`) F`

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

  @Currently no method is installed!@

- `IsPNilpotent(` G`,` p`) F`

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

  ## 37.16 Numerical Group Attributes

- `AbelianInvariants(` G`) A`

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

  (See IndependentGeneratorsOfAbelianGroup to obtain actual generators).

      gap> g:=Group((1,2,3,4),(1,2),(5,6));;
      gap> AbelianInvariants(g);
      [ 2, 2 ]

- `Exponent(` G`) A`

  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 ge=1 for all g ∈ G

      gap> Exponent(g);
      12

  Again the following are mathematical attributes, but not GAP `Attributes` as they are depending on a parameter:

- `EulerianFunction(` G`,` n`) O`

  returns the number of n-tuples (g1, g2, …gn) of elements of the group G that generate the whole group G. The elements of an n-tuple need not be different. If the Library of Tables of Marks (see Chapter Tables of Marks) covers the group G, you may also use `EulerianFunctionByTom` (see EulerianFunctionByTom).

      gap> EulerianFunction(g,2);
      432

  ## 37.17 Subgroup Series

  In group theory many subgroup series are considered, and GAP provides commands to compute them. In the following sections, there is always a series G = U1 > U2 > … > Um = 〈1 〉 of subgroups considered. A series also may stop without reaching G or 〈1〉.

  A series is called subnormal if every Ui+1 is normal in Ui.

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

  A series of normal subgroups is called central if Ui/Ui+1 is central in G/Ui+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`) A`

  is a series of normal subgroups of G which cannot be refined further. That is there is no normal subgroup N of G with Ui > N > Ui+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`) O`

  is a chief series of the group G going through the normal subgroups in the list l. l 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`) O`

  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`) O`

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

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

- `CompositionSeries(` G`) A`

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

- `DisplayCompositionSeries(` G`) F`

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

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

- `DerivedSeriesOfGroup(` G`) A`

  The derived series of a group is obtained by Ui+1=Ui′. It stops if Ui is perfect.

- `DerivedLength(` G`) A`

  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`) A`
- `ElementaryAbelianSeriesLargeSteps(` G`) A`
- `ElementaryAbelianSeries( [`G`,` NT1`,` NT2`, ...] ) A`

  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 is constructed.

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

- `InvariantElementaryAbelianSeries(` G`,` morph`[,` N`[,` fine`]] ) O`

  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`) A`

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

- `UpperCentralSeriesOfGroup(` G`) A`

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

- `PCentralSeries(` G`,` p`) F`

  The p-central series of G is defined by U1:=G, Ui:=[G,Ui−1]Ui−1p.

- `JenningsSeries(` G`) A`

  For a p-group G, this function returns its Jennings series. This series is defined by setting G1=G and for i ≥ 0, Gi+1=[Gi,G]Gjp, where j is the smallest integer ≥ i/p.

- `DimensionsLoewyFactors(` G`) A`

  This operation computes the dimensions of the factors of the Loewy series of G. (See Hup82, p. 157 for the slightly complicated definition of the Loewy Series.)

  The dimensions are computed via the `JenningsSeries` 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`) F`

  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.

- `IntermediateGroup(` G`,` U`) F`

  This routine tries to find a subgroup E of G, such that G > E > U. If U is maximal, it returns `fail`. This is done by finding minimal blocks for the operation of G on the right cosets of U.

- `IntermediateSubgroups(` G`,` U`) O`

  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 components `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 subgroup number i is a maximal subgroup of subgroup number j, the numbers 0 and 1+length(`subgroups`) are used to denote U and G respectively.

  ## 37.18 Factor Groups

- `NaturalHomomorphismByNormalSubgroup(` G`,` N`) F`
- `NaturalHomomorphismByNormalSubgroupNC(` G`,` N`) F`

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

- `FactorGroup(` G`,` N`) F`
- `FactorGroupNC(` G`,` N`) O`

  returns the image of the `NaturalHomomorphismByNormalSubgroup(`G`,`N`)`. The `NC` version does not test whether N is normal in G.

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

- `CommutatorFactorGroup(` G`) A`

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

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

- `MaximalAbelianQuotient(` grp`) A`

  returns an epimorphism from grp onto the maximal abelian quotient of grp. The kernel of this epimorphism is the derived subgroup.

- `HasAbelianFactorGroup(` G`,` N`) O`

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

- `HasElementaryAbelianFactorGroup(` G`,` N`) O`

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

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

- `CentralizerModulo(` G`,` N`,` elm`) O`

  Computes the full preimage of the centralizer CG/N(elm·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) ])

  ## 37.19 Sets of Subgroups

- `ConjugacyClassSubgroups(` G`,` U`) O`

  generates the conjugacy class of subgroups of G with representative U. This class is an external set, so functions such as `Representative`, (which returns U), `ActingDomain` (which returns G), `StabilizerOfExternalSet` (which returns the normalizer of U), and `AsList` work for it.

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

  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`) A`

  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). 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` (see Representative).

      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( [ (1,2)(3,4), (3,4) ] )^G, Group( [ (1,2)(3,4), (1,3,2,4) ] )^G,
        Group( [ (3,4), (2,4,3) ] )^G, Group( [ (1,3)(2,4), (1,4)(2,3), (1,2) ] )^G,
        Group( [ (1,3)(2,4), (1,4)(2,3), (2,4,3) ] )^G,
        Group( [ (1,3)(2,4), (1,4)(2,3), (2,4,3), (1,2) ] )^G ]

- `ConjugacyClassesMaximalSubgroups(` G`) A`

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

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

- `MaximalSubgroupClassReps(` G`) A`

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

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

- `MaximalSubgroups(` G`) A`

  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.

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

- `NormalSubgroups(` G`) A`

  returns a list of all normal subgroups of G.

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

  The algorithm used for the computation of normal subgroups of permutation groups and pc groups is described in Hulpke98.

- `MaximalNormalSubgroups(` G`) A`

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

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

- `MinimalNormalSubgroups(` G`) A`

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

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

  ## 37.20 Subgroup Lattice

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

- `LatticeSubgroups(` G`) A`

  computes the lattice of subgroups of the group G. This lattice has the conjugacy classes of subgroups as attribute `ConjugacyClassesSubgroups` (see ConjugacyClassesSubgroups) 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( [ (1,2)(3,4), (3,4) ] )^G, Group( [ (1,2)(3,4), (1,3,2,4) ] )^G,
        Group( [ (3,4), (2,4,3) ] )^G, Group( [ (1,3)(2,4), (1,4)(2,3), (1,2) ] )^G,
        Group( [ (1,3)(2,4), (1,4)(2,3), (2,4,3) ] )^G,
        Group( [ (1,3)(2,4), (1,4)(2,3), (2,4,3), (1,2) ] )^G ]

- `ClassElementLattice(` C`,` n`) O`

  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, `AsList(C)` can give a different arrangement of the class elements!

- `MaximalSubgroupsLattice(` lat`) A`

  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` 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 [cls,nr] which means that the nrst subgroup in class number cls is a maximal subgroup of the representative.

  The number nr corresponds to access via `ClassElementLattice` and not necessarily the `AsList` arrangement! See also MinimalSupergroupsLattice.

      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([ (1,2)(3,4), (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`) A`

  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` 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 [cls,nr] which means that the nrst subgroup in class number cls is a minimal supergroup of the representative.

  The number nr corresponds to access via `ClassElementLattice` and not necessarily the `AsList` arrangement! See also MaximalSubgroupsLattice.

      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

- `RepresentativesPerfectSubgroups(` G`) A`
- `RepresentativesSimpleSubgroups(` G`) A`

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

      gap> m11:=TransitiveGroup(11,6);
      M(11)
      gap> r:=RepresentativesPerfectSubgroups(m11);
      [ Group([ (3,5,8)(4,11,7)(6,9,10), (2,3)(4,10)(5,9)(8,11) ]),
        Group([ (1,2,4)(5,11,8)(6,9,7), (2,3)(4,10)(5,9)(8,11) ]),
        Group([ (3,4,10)(5,11,6)(7,9,8), (1,4,9)(3,6,10)(7,11,8) ]),
        Group([ (1,2,5)(3,11,10)(6,7,8), (2,3)(4,10)(5,9)(8,11) ]), M(11),
        Group(()) ]
      gap> List(r,Size);
      [ 60, 60, 360, 660, 7920, 1 ]

- `ConjugacyClassesPerfectSubgroups(` G`) A`

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

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

- `Zuppos(` G`) A`

  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 V`

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

  ## 37.21 Specific Methods for Subgroup Lattice Computations

- `LatticeByCyclicExtension(` G`[,` func`[,` noperf`]] ) F`

  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.

  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`] ) F`

  Let G be an elementary abelian group (that is a vector space) and homs 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 subspaces. If the optional parameter dims is given, only subspaces of this dimension are computed.

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

- `SubgroupsSolvableGroup(` G`[,` opt`] ) F`

  This function (implementing the algorithm published in Hulpke99) computes subgroups of a solvable group G, using the homomorphism principle. It returns a list of representatives up to G-conjugacy.

  The optional argument opt is a record, which may be used to put restrictions on the subgroups computed. The following record components of opt are recognized and have the following effects:

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

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

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

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

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

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

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

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

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

- `SizeConsiderFunction(` size`) F`

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

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

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

- `ExactSizeConsiderFunction(` size`) F`

  This function returns a function consider of four arguments that can be used in `SubgroupsSolvableGroup` (see SubgroupsSolvableGroup) for the option `consider` to compute subgroups whose sizes are exactly size.

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

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

- `InfoPcSubgroup V`

  Information function for the subgroup lattice functions using pcgs.

  ## 37.22 Special Generating Sets

- `GeneratorsSmallest(` G`) A`

  returns a ``smallest'' generating set for the group G. This is the lexicographically (using GAPs order of group elements) smallest list l of elements of G such that G=〈l〉 and li ∉ 〈l1,…,li−1〉 (in particular l1 is not the one 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`) A`

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

- `MinimalGeneratingSet(` G`) A`

  returns a generating set of G of minimal possible length.

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

- `SmallGeneratingSet(` G`) A`

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

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

- `IndependentGeneratorsOfAbelianGroup(` A`) A`

  returns a set of generators g of prime-power order of the abelian group A such that A is the direct product of the cyclic groups generated by the gi.

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

  ## 37.23 1-Cohomology

  Let G be a finite group and M an elementary abelian normal p-subgroup of G. Then the group of 1-cocycles Z1( G/M, M ) is defined as

  Z1(G/M, M) = { γ: G/M → M | ∀g1, g2 ∈ G : γ(g1 M ·g2 M ) = γ(g1 M)g2 ·γ(g2 M) }

  and is a GF(p)-vector space.

  The group of 1-coboundaries B1( G/M, M ) is defined as

  B1(G/M, M) = { γ: G/M → M | ∃m ∈ M ∀g ∈ G : γ(gM) = (m−1)g ·m }

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

  Let α be the isomorphism of M into a row vector space W and (g1,…,gl) representatives for a generating set of G/M. Then there exists a monomorphism β of Z1( G/M, M ) in the l-fold direct sum of W, such that β( γ) = ( α( γ(g1 M) ),…, α( γ(gl M) ) ) for every γ ∈ Z1( G/M, M ).

- `OneCocycles(` G`,` M`) O`
- `OneCocycles(` gens`,` M`) O`
- `OneCocycles(` G`,` mpcgs`) O`
- `OneCocycles(` gens`,` mpcgs`) O`

  Computes the group of 1-Cocycles Z1(G /M ,M ). The normal subgroup M may be given by a (Modulo)Pcgs mpcgs. In this case the whole calculation is performed modulo the normal subgroup defined by the `DenominatorOfModuloPcgs(`mpcgs`)` (see Polycyclic Generating Systems). Similarly the group G may instead be specified by a set of elements gens that are representatives for a generating system for the factor group G/M. If this is done the 1-Cocycles are computed with respect to these generators (otherwise the routines try to select suitable generators themselves).

- `OneCoboundaries(` G`,` M`) O`

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

  The operations `OneCocycles` 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 Z1. The vectors are represented in dimension a·b where a is the length of `generators` and pb the size of M.

  `oneCoboundaries`
    A space of row vectors that represents B1.

  `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 `Group`s in the sense of 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( oneCoboundaries := <vector space over GF(2), with 2 generators>,
        oneCocycles := <vector space over GF(2), with 2 generators>,
        generators := [ (3,4), (2,4,3) ], isSplitExtension := true,
        complement := Group([ (3,4), (2,4,3) ]),
        cocycleToList := function( c ) ... end,
        listToCocycle := function( L ) ... end,
        cocycleToComplement := function( c ) ... end,
        factorGens := [ (3,4), (2,4,3) ],
        complementToCocycle := function( K ) ... end )
      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)]);
      [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ]
      gap> oc.cocycleToComplement([ 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0 ]);
      Group([ (1,2), (1,2,3) ])
      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)));
      [ 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0 ]

  The factor group H1(G/M,M)=Z1(G/M,M)/B1(G/M,M) is called the first cohomology group. Currently there is no function which explicitly computes this group. The easiest way to represent it is as a vector space complement to B1 in Z1.

  If the only purpose of the calculation of H1 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`.

- `OCOneCocycles(` ocr`,` onlySplit`) O`

  is the more technical function to compute 1-cocycles. It takes an record ocr as first argument which must contain at least the components `group` for G and `modulePcgs` for a (modulo) pcgs of M. This record will also be returned with components as described under `OneCocycles` (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`, `OneCocyclesOC` returns `false` as soon as possible if the extension does not split.

- `ComplementclassesEA(` G`,` N`) O`

  computes `Complementclasses` to an elementary abelian normal subgroup N via 1-Cohomology. Normally, a user program should call `Complementclasses` (see Complementclasses) instead, which also works for a solvable (not necessarily elementary abelian) N.

- `InfoCoh V`

  The info class for the cohomology calculations is `InfoCoh`.

  ## 37.24 Schur Covers and Multipliers

- `EpimorphismSchurCover(` G`[,` pl`] ) O`

  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` 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`) O`

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

  At the moment this cover is represented as a finitely presented group and `IsomorphismPermGroup` would be needed to convert it to a permutation group.

  If also the relation to G is needed, `EpimorphismSchurCover` should be used.

      gap> g:=Group((1,2,3,4),(1,2));;
      gap> epi:=EpimorphismSchurCover(g);
      [ f1, f2, f3 ] -> [ (3,4), (2,4,3), (1,4)(2,3) ]
      gap> Size(Source(epi));
      48

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

  There is another operation which only returns the structure of the Multiplier.

- `AbelianInvariantsMultiplier(` G`) A`

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

      gap> AbelianInvariantsMultiplier(g);
      [ 2 ]

  Note that the following example will take some time.

      gap> AbelianInvariantsMultiplier(PSU(6,2));
      [ 2, 2, 3 ]

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

  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 a Schur Cover of G is invariant and isomorphic to the NonabelianExteriorSquare of G BJR87.

- `Epicentre(` G`) A`
- `ExteriorCentre(` G`) A`

  There are various ways of describing the epicentre of a group. It is the smallest normal subgroup N of G such that G/N is a central quotient of a group. It is also equal to the Exterior Center of G Ellis98.

- `NonabelianExteriorSquare(` G`) O`

  Computes the Nonabelian Exterior Square G∧G of a group G which for a finitely presented group is the derived subgroup of any Schur Cover of G BJR87.

- `EpimorphismNonabelianExteriorSquare(` G`) O`

  Computes the mapping G∧G → G. The kernel of this mapping is equal to the Schur Multiplicator of G.

- `IsCentralFactor(` G`) P`

  This method determines if there exists a group H such that G is isomormorphic 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 BFS79.

  ## 37.25 Tests for the Availability of Methods

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

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

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

- `CanEasilyTestMembership(` grp`) F`

  This filter indicates whether a group can test membership of elements in grp (via the operation `in`) in reasonable time. It is used by the method selection to decide whether an algorithm that relies on membership tests may be used.

- `CanComputeSize(` dom`) F`

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

- `CanComputeSizeAnySubgroup(` grp`) F`

  This filter indicates whether grp can easily compute the size of any subgroup. (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`) F`

  This filter indicates whether the index [G:H] (which might be `infinity`) can be computed. It assumes that H ≤ G. (see CanComputeIsSubset)

- `CanComputeIsSubset(` A`,` B`) O`

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

- `KnowsHowToDecompose(` G`) P`
- `KnowsHowToDecompose(` G`,` gens`) O`

  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 `GeneratorsOfGroup`.

  This property can be used for example to check whether a `GroupHomomorphismByImages` can be reasonably defined from this group.

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  GAP 4 manual
  December 2008