# 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.

GAP 3.4.4
April 1997