The group consists of the residue classes of the integers modulo under addition. This Demonstration illustrates the action of a group homomorphism . Such a mapping must have the form for some in . The homomorphism is an automorphism if and only if is a generator of the cyclic group .

The operation for an Abelian group is typically written as addition. A group homomorphism is a mapping from a group to a group such that for all and in . The kernel of a homomorphism consists of those elements in whose image is the identity in . An isomorphism is a one-to-one homomorphism from onto . An automorphism of a group is an isomorphism .