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) ] )
G.operations.ConjugationGroupHomomorphism( G, H, x )
and returns that value.
The default function called is
It just creates a homomorphism record with range G, source H, and the
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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