Groups and homomorphisms
Functions for groups This method has been transferred from package &ResClasses;.

It provides a method for Comm when the argument is a list (enclosed in square brackets), and calls the function LeftNormedComm.

gap> Comm( [ (1,2), (2,3) ] ); (1,2,3) gap> Comm( [(1,2),(2,3),(3,4),(4,5),(5,6)] ); (1,5,6) gap> Comm(Comm(Comm(Comm((1,2),(2,3)),(3,4)),(4,5)),(5,6)); ## the same (1,5,6)
This function has been transferred from package &ResClasses;.

It tests whether two elements in a group commute.

gap> D12 := DihedralGroup( 12 ); <pc group of size 12 with 3 generators> gap> SetName( D12, "D12" ); gap> a := D12.1;; b := D12.2;; gap> IsCommuting( a, b ); false
This function has been transferred from package &RCWA;.

The operation ListOfPowers(g,exp) returns the list [g,g^2,...,g^{exp}] of powers of the element g.

gap> ListOfPowers( 2, 20 ); [ 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576 ] gap> ListOfPowers( (1,2,3)(4,5), 12 ); [ (1,2,3)(4,5), (1,3,2), (4,5), (1,2,3), (1,3,2)(4,5), (), (1,2,3)(4,5), (1,3,2), (4,5), (1,2,3), (1,3,2)(4,5), () ] gap> ListOfPowers( D12.2, 6 ); [ f2, f3, f2*f3, f3^2, f2*f3^2, <identity> of ... ]
This function has been transferred from package &RCWA;.

This operation returns a list containing the generators of G followed by the inverses of these generators.

gap> GeneratorsAndInverses( D12 ); [ f1, f2, f3, f1, f2*f3^2, f3^2 ] gap> GeneratorsAndInverses( SymmetricGroup(5) ); [ (1,2,3,4,5), (1,2), (1,5,4,3,2), (1,2) ]
These three functions have been transferred from package &ResClasses;.

Fitting series The upper and lower Fitting series and the Fitting length of a solvable group are described here: https://en.wikipedia.org/wiki/Fitting_length.

gap> upp:= UpperFittingSeries( D12 );; gap> List( upp, StructureDescription ); [ "1", "C6", "D12" ] gap> low:= LowerFittingSeries( D12 );; gap> List( low, StructureDescription ); [ "D12", "C3", "1" ] gap> FittingLength( D12 ); 2 gap> S4 := SymmetricGroup( 4 );; gap> UpperFittingSeries( S4 ); [ Group(()), Group([ (1,2)(3,4), (1,4)(2,3) ]), Group([ (2,4,3), (1,2)(3,4) ]), Group([ (3,4), (2,4,3), (1,2)(3,4) ]) ] gap> List( last, StructureDescription ); [ "1", "C2 x C2", "A4", "S4" ] gap> LowerFittingSeries( S4 ); [ Sym( [ 1 .. 4 ] ), Alt( [ 1 .. 4 ] ), Group([ (1,4)(2,3), (1,2)(3,4) ]), Group(()) ] gap> List( last, StructureDescription ); [ "S4", "A4", "C2 x C2", "1" ] gap> FittingLength( S4); 3
Left Cosets for Groups Since &GAP; uses right actions by default, the library contains the operation RightCoset(U,g) for constructing the right coset Ug of a subgroup U \leq G and an element g \in G. It has been noted in the reference manual that, by inverting all the elements in Ug, the left coset g^{-1}U is obtained.

Just for the sake of completeness, from August 2022 this package provides the operation LeftCoset(g,U) for constructing the left coset gU. Users are strongly recommended to continue to use RightCoset for all serious calculations, since left cosets have a much simpler implementation and do not behave exactly like right cosets.

The methods for left cosets which are provided generally work by converting gU to Ug^{-1}; applying the equivalent method for right cosets; and, if necessary, converting back again to left cosets.

G acts on left cosets by OnLeftInverse: (gU)^{g_0} = g_0^{-1}*(gU) = (g_0^{-1}g)U.

gap> a4 := Group( (1,2,3), (2,3,4) );; SetName( a4, "a4" ); gap> k4 := Group( (1,2)(3,4), (1,3)(2,4) );; SetName( k4, "k4" ); gap> rc := RightCosets( a4, k4 ); [ RightCoset(k4,()), RightCoset(k4,(2,3,4)), RightCoset(k4,(2,4,3)) ] gap> lc := LeftCosets( a4, k4 ); [ LeftCoset((),k4), LeftCoset((2,4,3),k4), LeftCoset((2,3,4),k4) ] gap> AsSet( lc[2] ); [ (2,4,3), (1,2,3), (1,3,4), (1,4,2) ] gap> LeftCoset( (1,4,2), k4 ) = lc[2]; true gap> Representative( lc[2] ); (2,4,3) gap> ActingDomain( lc[2] ); k4 gap> (1,4,3) in lc[3]; true gap> (1,2,3)*lc[2] = lc[3]; true gap> lc[2]^(1,3,2) = lc[3]; true
Inverse The inverse of the left coset gU is the right coset Ug^{-1}, and conversely. This is an abuse of the attribute Inverse, since the standard requirement, that x*x^{-1} is an identity, does not hold.

gap> Inverse( rc[3] ) = lc[3]; true gap> Inverse( lc[2] ) = rc[2]; true
Functions for group homomorphisms This function has been transferred from package &RCWA;.

It constructs a group homomorphism which maps the generators of G to those of H. Its intended use is when G is a free group, and a warning is printed when this is not the case. Note that anything may happen if the resulting map is not a homomorphism!

gap> G := Group( (1,2,3), (3,4,5), (5,6,7), (7,8,9) );; gap> phi := EpimorphismByGenerators( FreeGroup("a","b","c","d"), G ); [ a, b, c, d ] -> [ (1,2,3), (3,4,5), (5,6,7), (7,8,9) ] gap> PreImagesRepresentativeNC( phi, (1,2,3,4,5,6,7,8,9) ); d*c*b*a gap> a := G.1;; b := G.2;; c := G.3;; d := G.4;; gap> d*c*b*a; (1,2,3,4,5,6,7,8,9) gap> ## note that it is easy to produce nonsense: gap> epi := EpimorphismByGenerators( Group((1,2,3)), Group((8,9)) ); Warning: calling GroupHomomorphismByImagesNC without checks [ (1,2,3) ] -> [ (8,9) ] gap> IsGroupHomomorphism( epi ); true gap> Image( epi, (1,2,3) ); () gap> Image( epi, (1,3,2) ); (8,9)
If \phi_1 : G_1 \to H and \phi_2 : G_2 \to H are two group homomorphisms with the same range, then their pullback is the subgroup of G_1 \times G_2 consisting of those elements (g_1,g_2) such that \phi_1 g_1 = \phi_2 g_2.

The attribute PullbackInfo of a pullback group P is similar to DirectProductInfo for a direct product of groups. Its value is a record with the following components:

directProduct the direct product G_1 \times G_2, and projections a list with the two projections onto G_1 and G_2. There are no embeddings in this record, but it is possible to use the embeddings into the direct product, see .

gap> s4 := Group( (1,2),(2,3),(3,4) );; gap> s3 := Group( (5,6),(6,7) );; gap> c3 := Subgroup( s3, [ (5,6,7) ] );; gap> f := GroupHomomorphismByImages( s4, s3, > [(1,2),(2,3),(3,4)], [(5,6),(6,7),(5,6)] );; gap> i := GroupHomomorphismByImages( c3, s3, [(5,6,7)], [(5,6,7)] );; gap> Pfi := Pullback( f, i ); Group([ (2,3,4)(5,7,6), (1,2)(3,4) ]) gap> StructureDescription( Pfi ); "A4" gap> info := PullbackInfo( Pfi ); rec( directProduct := Group([ (1,2), (2,3), (3,4), (5,6,7) ]), projections := [ [ (2,3,4)(5,7,6), (1,2)(3,4) ] -> [ (2,3,4), (1,2)(3,4) ], [ (2,3,4)(5,7,6), (1,2)(3,4) ] -> [ (5,7,6), () ] ] ) gap> g := (1,2,3)(5,6,7);; gap> ImageElm( info!.projections[1], g ); (1,2,3) gap> ImageElm( info!.projections[2], g ); (5,6,7) gap> dp := info!.directProduct;; gap> a := ImageElm( Embedding( dp, 1 ), (1,4,3) );; gap> b := ImageElm( Embedding( dp, 2 ), (5,7,6) );; gap> a*b in Pfi; true
This function was added by Thomas Breuer, following discussions with Hongyi Zhao (see https://github.com/gap-packages/hap/issues/73).

Let G1 and G2 be two groups, Z1 be a central subgroup of G1, and Phi be an isomorphism from Z1 to a central subgroup of G2. The central product defined by these arguments is the factor group of the direct product of G1 and G2 by the central subgroup \{ (z, (Phi(z))^{-1}) : z \in Z1 \}.

The attribute of a group G that has been created by is similar to for pullback groups. Its value is a record with the following components.

projection the epimorphism from the direct product of G1 and G2 to G, and phi the map Phi. Note that one can access the direct product as the value of the projection map, and one can access G1 and G2 as the two embeddings of this direct product, see .
gap> g1 := DihedralGroup( 8 ); <pc group of size 8 with 3 generators> gap> c1 := Centre( g1 ); Group([ f3 ]) gap> cp1 := CentralProduct( g1, g1, c1, IdentityMapping( c1 ) ); Group([ f1, f2, f5, f3, f4, f5 ]) gap> IsomorphismGroups( cp1, ExtraspecialGroup( 2^5, "+" ) ) <> fail; true gap> g2 := QuaternionGroup( 8 ); <pc group of size 8 with 3 generators> gap> c2 := Centre( g2 ); Group([ y2 ]) gap> cp2 := CentralProduct( g2, g2, c2, IdentityMapping( c2 ) ); Group([ f1, f2, f5, f3, f4, f5 ]) gap> IsomorphismGroups( cp1, ExtraspecialGroup( 2^5, "+" ) ) <> fail; true gap> info2 := CentralProductInfo( cp2 ); rec( phi := IdentityMapping( Group([ y2 ]) ), projection := [ f1, f2, f3, f4, f5, f6 ] -> [ f1, f2, f5, f3, f4, f5 ] ) gap> Source( Embedding( Source( info2.projection ), 1 ) ) = g2; true
An endomorphism f\colon G \to G is idempotent if f^2 = f. It has an image R \leq G, is the identity map when restricted to R, and has a kernel N which has trivial intersection with R and has size |G|/|R|.

The operation IdempotentEndomorphismsWithImage(genG,R) returns a list of the images of the generating set genG of a group G under the idempotent endomorphisms with image R.

The attribute IdempotentEndomorphismsData(G) returns a record data with fields data.gens, a fixed generating set for G, and data.images a list of the non-empty outputs of IdempotentEndomorphismsWithImage(genG,R) obtained by iterating over all subgroups R of G.

The operation IdempotentEndomorphisms(G) returns the list of these mappings obtained using IdempotentEndomorphismsData(G). The first of these is the zero map, the last is the identity.

gap> gens := [ (1,2,3,4), (1,2)(3,4) ];; gap> d8 := Group( gens );; gap> SetName( d8, "d8" ); gap> c2 := Subgroup( d8, [ (2,4) ] );; gap> SortedList( IdempotentEndomorphismsWithImage( gens, c2 ) ); [ [ (), (2,4) ], [ (2,4), () ] ] gap> data:= IdempotentEndomorphismsData( d8 );; gap> data.images[1]; [ [ (), () ] ] gap> List( data.images, Length ); [ 1, 2, 2, 2, 2, 1 ] gap> all:= IdempotentEndomorphisms( d8 );; gap> Length( all ); 10 gap> all[1]; [ (1,2,3,4), (1,2)(3,4) ] -> [ (), () ] gap> Size( Image( all[1] ) ); 1 gap> Last( all ) = IdentityMapping( d8 ); true
The quaternion group q8 is an example of a group with a tail: there is only one subgroup in the lattice which covers the identity subgroup. The only idempotent isomorphisms of such groups are the identity mapping and the zero mapping because the only pairs N,R are the whole group and the identity subgroup.
gap> q8 := QuaternionGroup( 8 );; gap> IdempotentEndomorphisms( q8 ); [ [ x, y ] -> [ <identity> of ..., <identity> of ... ], [ x, y ] -> [ x, y ] ]
Given group homomorphisms f_1 : G_1 \to G_2 and f_2 : H_1 \to H_2, this operation return the product homomorphism f_1 \times f_2 : G_1 \times G_2 \to H_1 \times H_2.

gap> c4 := Group( (1,2,3,4) );; gap> c2 := Group( (5,6) );; gap> f1 := GroupHomomorphismByImages( c4, c2, [(1,2,3,4)], [(5,6)] );; gap> c3 := Group( (1,2,3) );; gap> c6 := Group( (1,2,3,4,5,6) );; gap> f2 := GroupHomomorphismByImages( c3, c6, [(1,2,3)], [(1,3,5)(2,4,6)] );; gap> c4c3 := DirectProduct( c4, c3 ); Group([ (1,2,3,4), (5,6,7) ]) gap> c2c6 := DirectProduct( c2, c6 ); Group([ (1,2), (3,4,5,6,7,8) ]) gap> f := DirectProductOfFunctions( c4c3, c2c6, f1, f2 ); [ (1,2,3,4), (5,6,7) ] -> [ (1,2), (3,5,7)(4,6,8) ] gap> ImageElm( f, (1,4,3,2)(5,7,6) ); (1,2)(3,7,5)(4,8,6)
Let A_1,A_2 be groups of automorphism of groups G_1,G_2 respectively. The output of this function is a group A_1 \times A_2 of automorphisms of G_1 \times G_2.

gap> c9 := Group( (1,2,3,4,5,6,7,8,9) );; gap> ac9 := AutomorphismGroup( c9 );; gap> q8 := QuaternionGroup( IsPermGroup, 8 );; gap> aq8 := AutomorphismGroup( q8 );; gap> A := DirectProductOfAutomorphismGroups( ac9, aq8 ); <group with 5 generators> gap> genA := GeneratorsOfGroup( A );; gap> G := Source( genA[1] ); Group([ (1,2,3,4,5,6,7,8,9), (10,14,12,16)(11,17,13,15), (10,11,12,13) (14,15,16,17) ]) gap> a := genA[1]*genA[5]; [ (1,2,3,4,5,6,7,8,9), (10,14,12,16)(11,17,13,15), (10,11,12,13)(14,15,16,17) ] -> [ (1,3,5,7,9,2,4,6,8), (10,16,12,14)(11,15,13,17), (10,11,12,13)(14,15,16,17) ] gap> ImageElm( a, (1,9,8,7,6,5,4,3,2)(10,14,12,16)(11,17,13,15) ); (1,8,6,4,2,9,7,5,3)(10,16,12,14)(11,15,13,17)