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<!-- This is an automatically generated file. --> <Chapter Label="Chapter_Group_homomorphisms"> <Heading>Group homomorphisms</Heading> <Section Label="Chapter_Group_homomorphisms_Section_Representatives_of_homomorphis <Heading>Representatives of homomorphisms between groups</Heading> Please note that the functions below are only implemented for finite groups. <ManSection> <Func Arg="G" Name="RepresentativesAutomorphismClasses" /> <Returns>a list of the automorphisms of <A>G</A> up to composition with inner automorphisms. </Returns> <Description> <P/> </Description> </ManSection> <ManSection> <Func Arg="G" Name="RepresentativesEndomorphismClasses" /> <Returns>a list of the endomorphisms of <A>G</A> up to composition with inner automorphisms. </Returns> <Description> This does the same as calling <C>AllHomomorphismClasses(<A>G</A>,<A>G</A>)</C>, but should be faster for abelian and non-2-generated groups. For 2-generated groups, this function behaves nearly identical to <Ref Func="AllHomomorphismClasses" BookName="Ref" Style="Number"/>. </Description> </ManSection> <ManSection> <Func Arg="H, G" Name="RepresentativesHomomorphismClasses" /> <Returns>a list of the homomorphisms from <A>H</A> to <A>G</A>, up to composition with inner automorphis </Returns> <Description> This does the same as calling <C>AllHomomorphismClasses(<A>H</A>,<A>G</A>)</C>, but should be faster for abelian and non-2-generated groups. For 2-generated groups, this function behaves nearly identical to <Ref Func="AllHomomorphismClasses" BookName="Ref" Style="Number"/>. </Description> </ManSection> <Example><![CDATA[ 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 ]]></Example> </Section> <Section Label="Chapter_Group_homomorphisms_Section_Coincidence_and_fixed_point_gr <Heading>Coincidence and fixed point groups</Heading> <ManSection> <Func Arg="endo" Name="FixedPointGroup" /> <Returns>the subgroup of <C>Source(<A>endo</A>)</C> consisting of the elements fixed under the endomo </Returns> <Description> <P/> </Description> </ManSection> <ManSection> <Func Arg="hom1, hom2[, ...]" Name="CoincidenceGroup" /> <Returns>the subgroup of <C>Source(<A>hom1</A>)</C> consisting of the elements <C>h</C> for which <C>h^<A>hom ... </Returns> <Description> For infinite non-abelian groups, this function relies on a mixture of the algorithms described in <Cite Key='roma16-a' Where='Thm. 2'/>, <Cite Key='bkl20-a' Where='Sec. 5.4'/> and <Cite Key='roma21-a' Where='Sec. 7'/>. </Description> </ManSection> <Example><![CDATA[ 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 ]]></Example> </Section> <Section Label="Chapter_Group_homomorphisms_Section_Induced_and_restricted_group_h <Heading>Induced and restricted group homomorphisms</Heading> <ManSection> <Func Arg="epi1, epi2, hom" Name="InducedHomomorphism" /> <Returns>the homomorphism induced by <A>hom</A> between the images of <A>epi1</A> and <A>epi2</A>. </Returns> <Description> Let <A>hom</A> be a group homomorphism from a group <C>H</C> to a group <C>G</C>, let <A>epi1</A> be an epimorphism from <C>H</C> to a group <C>Q</C> and let <A>epi2</A> be an epimorphism from <C>G</C> to a group <C>P</C> such that the kernel of <A>epi1</A> is mapped into the kernel of <A>epi2</A> by <A>hom</A>. This command returns the homomorphism from <C>Q</C> to <C>P</C> that maps <C>h^<A>epi1</A></C> to <C>(h^<A>hom</A>)^<A>epi2</A></C>, for any element <C>h</C> of <C>H</C>. This function generalises <Ref Func="InducedAutomorphism" BookName="ref" Style="Number"/> to homomorphisms. </Description> </ManSection> <ManSection> <Func Arg="hom, N, M" Name="RestrictedHomomorphism" /> <Returns>the homomorphism <A>hom</A>, but restricted as a map from <A>N</A> to <A>M</A>. </Returns> <Description> Let <A>hom</A> be a group homomorphism from a group <C>H</C> to a group <C>G</C>, and let <A>N</A> be subgroup of <C>H</C> such that its image under <A>hom</A> is a subgroup of <A>M</A>. This command returns the homomorphism from <A>N</A> to <A>M</A> that maps <C>n</C> to <C>n^<A>hom</A></C> for any element <C>n</C> of <A>N</A>. No checks are made to verify that <A>hom</A> maps <A>N</A> into <A>M</A>. This function is similar to <Ref Func="RestrictedMapping" BookName="ref" Style="Number"/>, but its range is explicitly set to <A>M</A>. </Description> </ManSection> <Example><![CDATA[ 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 ]]></Example> </Section> </Chapter> ¤ Dauer der Verarbeitung: 0.23 Sekunden (vorverarbeitet) ¤*© Formatika GbR, Deutschland |
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2026-03-28