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4 Homomorphisms

4.1 Representatives of homomorphisms between groups

Please note that the functions below are only implemented for finite groups.

4.1-1 RepresentativesAutomorphismClasses

Let G be a group. This command returns a list of the automorphisms of G up to composition with inner automorphisms.

4.1-2 RepresentativesEndomorphismClasses

Let G be a group. This command returns a list of the endomorphisms of G up to composition with inner automorphisms. This does the same as calling AllHomomorphismClasses(G,G), but should be faster for abelian and non-2-generated groups. For 2-generated groups, this function takes its source code from AllHomomorphismClasses.

4.1-3 RepresentativesHomomorphismClasses

Let G and H be groups. This command returns a list of the homomorphisms from H to G, up to composition with inner automorphisms of G. This does the same as calling AllHomomorphismClasses(H,G), but should be faster for abelian and non-2-generated groups. For 2-generated groups, this function takes its source code from AllHomomorphismClasses.

gap> G := AlternatingGroup( 6 );;
gap> Auts := RepresentativesAutomorphismClasses( G );;
gap> Size( Auts );
4
gap> ForAll( Auts, IsGroupHomomorphism and IsEndoMapping and IsBijective );
true
gap> Ends := RepresentativesEndomorphismClasses( G );;
gap> Size( Ends );
5
gap> ForAll( Ends, IsGroupHomomorphism and IsEndoMapping );
true
gap> H := SymmetricGroup( 5 );;
gap> Homs := RepresentativesHomomorphismClasses( H, G );;
gap> Size( Homs );
2
gap> ForAll( Homs, IsGroupHomomorphism );
true

4.2 Coincidence and Fixed Point Groups

4.2-1 FixedPointGroup

Let endo be an endomorphism of a group G. This command returns the subgroup of G consisting of the elements fixed under the endomorphism endo.

This function does the same as CoincidenceGroup(endo,\mathrm{id}_G).

4.2-2 CoincidenceGroup

Let hom1, hom2, ... be group homomorphisms from a group H to a group G. This command returns the subgroup of H consisting of the elements h for which h^hom1 = h^hom2 = ...

For infinite non-abelian groups, this function relies on a mixture of the algorithms described in [Rom16, Theorem 2], [BKL+20, Section 5.4] and [Rom21, Section 7].

gap> phi := GroupHomomorphismByImages( G, G, [ (1,2,3,4,5), (4,5,6) ],
>  [ (1,2,6,3,5), (1,4,5) ] );;
gap> FixedPointGroup( phi );
Group([ (1,2,6,4,3) ])
gap> psi := GroupHomomorphismByImages( H, G, [ (1,2)(3,5,4), (2,3)(4,5) ],
>  [ (1,4)(3,6), () ] );;
gap> khi := GroupHomomorphismByImages( H, G, [ (1,2)(3,5,4), (2,3)(4,5) ],
>  [ (1,2)(3,4), () ] );;
gap> CoincidenceGroup( psi, khi );
Group([ (1,2,3,4,5), (1,3,4,5,2) ])

4.3 Induced and restricted group homomorphisms

4.3-1 InducedHomomorphism

Let hom be a group homomorphism from a group H to a group G, let epi1 be an epimorphism from H to a group Q and let epi2 be an epimorphism from G to a group P such that the kernel of epi1 is mapped into the kernel of epi2 by hom. This command returns the homomorphism from Q to P induced by hom via epi1 and epi2, that is, the homomorphism from Q to P which maps h^epi1 to (h^hom)^epi2, for any element h of H. This generalises InducedAutomorphism to homomorphisms.

4.3-2 RestrictedHomomorphism

Let hom be a group homomorphism from a group H to a group G, and let N be subgroup of H such that its image under hom is a subgroup of M. This command returns the homomorphism from N to M induced by hom. This is similar to RestrictedMapping, but the range is explicitly set to M.

gap> G := ExamplesOfSomePcpGroups( 5 );;
gap> phi := GroupHomomorphismByImages( G, G, [ G.1, G.2, G.3, G.4 ],
>  [ G.1*G.4^-1, G.3, G.2*(G.3*G.4)^2, G.4^-1  ] );;
gap> N := DerivedSubgroup( G );;
gap> p := NaturalHomomorphismByNormalSubgroup( G, N );
[ g1, g2, g3, g4, g2^2, g3^2, g4^2 ] -> [ g1, g2, g3, g4, id, id, id ]
gap> ind := InducedHomomorphism( p, p, phi );
[ g1, g2, g3 ] -> [ g1*g4, g3, g2 ]
gap> Source( ind ) = Range( p ) and Range( ind ) = Range( p );
true
gap> res := RestrictedHomomorphism( phi, N, N );
[ g2^2, g3^2, g4^2 ] -> [ g3^2, g2^2*g3^4*g4^8, g4^-2 ]
gap> Source( res ) = N and Range( res ) = N;
true

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