KernelGroupHomomorphism( hom )
KernelGroupHomomorphism returns the kernel of the group homomorphism
hom as a subgroup of the group hom.source.
The kernel of a group homomorphism hom is the subset of elements x of the source G that are mapped to the identity of the range H, i.e., x^{hom} = H.identity.
gap> s4 := Group( (1,2), (1,2,3,4) );;
gap> v4 := Subgroup( s4, [ (1,2)(3,4), (1,3)(2,4) ] );;
gap> phi := NaturalHomomorphism( s4, s4/v4 );;
gap> KernelGroupHomomorphism( phi );
Subgroup( Group( (1,2), (1,2,3,4) ), [ (1,2)(3,4), (1,3)(2,4) ] )
gap> Kernel( phi );
Subgroup( Group( (1,2), (1,2,3,4) ), [ (1,2)(3,4), (1,3)(2,4) ] )
# since the source is a group this is equivalent to the above
gap> rho := GroupHomomorphismByImages( s4, Group( (1,2) ),
> [ (1,2), (1,2,3,4) ], [ (1,2), (1,2) ] );;
gap> Kernel( rho );
Subgroup( Group( (1,2), (1,2,3,4) ), [ (2,4,3), (1,4,3) ] )
KernelGroupHomomorphism first tests if hom.kernelGroupHomomorphism
is bound. If it is bound, KernelGroupHomomorphisms returns that value.
Otherwise it calls
hom.operations.KernelGroupHomomorphism( hom ), remembers the
returned value in hom.kernelGroupHomomorphism, and returns it.
The default function for this is MappingOps.KernelGroupHomomorphism,
which simply tries random elements of the source of hom, until the
subgroup generated by those that map to the identity has the correct
size, i.e., Size( hom.source ) / Size( Image( hom ) ). Note that
this implies that the image of hom and its size are computed. Look
under Kernel in the index to see for which group homomorphisms this
function is overlaid.
GAP 3.4.4