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8 Group homomorphisms

8.1 Representatives of homomorphisms between groups

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

8.1-1 RepresentativesAutomorphismClasses

Returns: a list of the automorphisms of G up to composition with inner automorphisms.

8.1-2 RepresentativesEndomorphismClasses

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 behaves nearly identical to AllHomomorphismClasses (Ref 40.9-2).

8.1-3 RepresentativesHomomorphismClasses

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 behaves nearly identical to AllHomomorphismClasses (Ref 40.9-2).

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

8.2 Coincidence and fixed point groups

8.2-1 FixedPointGroup

Returns: the subgroup of Source(endo) consisting of the elements fixed under the endomorphism endo.

8.2-2 CoincidenceGroup

Returns: the subgroup of Source(hom1) 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, Thm. 2], [BKL+20, Sec. 5.4] and [Rom21, Sec. 7].

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

8.3 Induced and restricted group homomorphisms

8.3-1 InducedHomomorphism

Returns: the homomorphism induced by hom between the images of epi1 and epi2.

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 that maps h^epi1 to (h^hom)^epi2, for any element h of H. This function generalises InducedAutomorphism (ref 40.7-7) to homomorphisms.

8.3-2 RestrictedHomomorphism

Returns: the homomorphism hom, but restricted as a map from N to M.

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 that maps n to n^hom for any element n of N. No checks are made to verify that hom maps N into M. This function is similar to RestrictedMapping (ref 32.2-13), but its range is explicitly set to M.

gap> G := PcGroupCode( 1018013, 28 );;
gap> phi := GroupHomomorphismByImages( G, G, [ G.1, G.3 ],
>  [ G.1*G.2*G.3^2, G.3^4 ] );;
gap> N := DerivedSubgroup( G );;
gap> p := NaturalHomomorphismByNormalSubgroup( G, N );
[ f1, f2, f3 ] -> [ f1, f2, <identity> of ... ]
gap> ind := InducedHomomorphism( p, p, phi );
[ f1 ] -> [ f1*f2 ]
gap> Source( ind ) = Range( p ) and Range( ind ) = Range( p );
true
gap> res := RestrictedHomomorphism( phi, N, N );
[ f3 ] -> [ f3^4 ]
gap> Source( res ) = N and Range( res ) = N;
true

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