7.110 ConjugationGroupHomomorphism

ConjugationGroupHomomorphism( G, H, x )

ConjugationGroupHomomorphism returns the homomorphism from G into H that takes each element g in G to the element g ^ x. G and H must have a common parent group P and x must lie in this parent group. Of course G ^ x must be a subgroup of H.

    gap> d12 := Group( (1,2,3,4,5,6), (2,6)(3,5) );; d12.name := "d12";;
    gap> c2 := Subgroup( d12, [ (2,6)(3,5) ] );
    Subgroup( d12, [ (2,6)(3,5) ] )
    gap> v4 := Subgroup( d12, [ (1,2)(3,6)(4,5), (1,4)(2,5)(3,6) ] );
    Subgroup( d12, [ (1,2)(3,6)(4,5), (1,4)(2,5)(3,6) ] )
    gap> x := ConjugationGroupHomomorphism( c2, v4, (1,3,5)(2,4,6) );
    ConjugationGroupHomomorphism( Subgroup( d12, 
    [ (2,6)(3,5) ] ), Subgroup( d12, [ (1,2)(3,6)(4,5), (1,4)(2,5)(3,6) 
     ] ), (1,3,5)(2,4,6) )
    gap> IsSurjective( x );
    false
    gap> Image( x );
    Subgroup( d12, [ (1,5)(2,4) ] ) 

ConjugationGroupHomomorphism calls
G.operations.ConjugationGroupHomomorphism( G, H, x ) and returns that value.

The default function called is GroupOps.ConjugationGroupHomomorphism. It just creates a homomorphism record with range G, source H, and the component element with the value x. It computes the image of an element g of G as g ^ x. If the sizes of the range and the source are equal the inverse of such a homomorphism is computed as a conjugation homomorphism from H to G by x^-1. To multiply two such homomorphisms their elements are multiplied. Look under ConjugationGroupHomomorphism in the index to see for which groups this default function is overlaid.

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