# 7.105 Group Homomorphisms

Since groups is probably the most important category of domains in GAP group homomorphisms are probably the most important homomorphisms (see chapter Homomorphisms)

A group homomorphism phi is a mapping that maps each element of a group G, called the source of phi, to an element of another group H, called the range of phi, such that for each pair x, y in G we have (xy)^phi = x^phi y^phi.

Examples of group homomorphisms are the natural homomorphism of a group into a factor group (see NaturalHomomorphism) and the homomorphism of a group into a symmetric group defined by an operation (see OperationHomomorphism). Look under group homomorphisms in the index for a list of all available group homomorphisms.

Since group homomorphisms are just a special case of homomorphisms, all functions described in chapter Homomorphisms are applicable to all group homomorphisms, e.g., the function to test if a homomorphism is an automorphism (see IsAutomorphism). More general, since group homomorphisms are just a special case of mappings all functions described in chapter Mappings are also applicable, e.g., the function to compute the image of an element under a group homomorphism (see Image).

The following sections describe the functions that test whether a mapping is a group homomorphism (see IsGroupHomomorphism), compute the kernel of a group homomorphism (see KernelGroupHomomorphism), how the general Mapping Functions for Group Homomorphisms), the natural homomorphism of a group onto a factor group (see NaturalHomomorphism), homomorphisms by conjugation (see ConjugationGroupHomomorphism, InnerAutomorphism), and the most general group homomorphism, which is defined by simply specifying the images of a set of generators (see GroupHomomorphismByImages).

GAP 3.4.4
April 1997