Definition 3

Let $G$ and $G'$ be groups with operators in $\Omega$. A homomorphism of groups with operators of $G$ into $G'$ is a homomorphism of the group $G$ into the group $G'$ such that

for all $\alpha\in\Omega$ and all $x\in G$.

An endomorphism of the group with operators $G$ is an endomorphism of the group $G$ which is permutable with all the homotheties of $G$.

As two homotheties of a group with operators $G$ are not necessarily permutable, a homothety of $G$ is not in general an endomorphism of the group with operators $G$.

The identity mapping of a group with operators is a homomorphism of groups with operators; the composition of two homomorphisms of groups with operators is also one. For a mapping to be an isomorphism of groups with operators, it is necessary and sufficient that it be a bijective homomorphism of groups with operators and the inverse mapping is then an isomorphism of groups with operators.

More generally, let G (resp. G') be a group with operators in $\Omega$ (resp. $\Omega'$). Let $\phi$ be a mapping of $\Omega$ into $\Omega'$. A $\phi$-homomorphism of G into G' is a homomorphism of the group G into the group G' such that

for all $\alpha \in \Omega$ and all $x \in G$.

In the rest of this paragraph we shall be given a set $\Omega$. Unless otherwise mentioned the groups with operators considered will admit $\Omega$ as set of operators.