# 43 Homomorphisms

An important special class of mappings are homomorphisms.

A mapping map is a homomorphism if the source and the range are domains of the same category, and map respects their structure. For example, if both source and range are groups and for each x,y in the source (xy)^{map} = x^{map} y^{map}, then map is a group homomorphism.

Homomorphism are created by homomorphism constructors, which are ordinary GAP functions that return homomorphisms, such as `FrobeniusAutomorphism` (see FrobeniusAutomorphism) or `NaturalHomomorphism` (see NaturalHomomorphism).

The first section in this chapter describes the function that tests whether a mapping is a homomorphism (see IsHomomorphism). The next sections describe the functions that test whether a homomorphism has certain properties (see IsMonomorphism, IsEpimorphism, IsIsomorphism, IsEndomorphism, and IsAutomorphism). The last section describes the function that computes the kernel of a homomorphism (see Kernel).

Because homomorphisms are just a special case of mappings all operations and functions described in chapter Mappings are applicable to homomorphisms. For example, the image of an element elm under a Operations for Mappings).

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GAP 3.4.4
April 1997