Definition 2

Let $\Omega$ be a set. A group $G$ together with an action of $\Omega$ on $G$ which is distributive with respect to the group law, is called a group with operators in $\Omega$.

In what follows $x^\alpha$ will denote the composition of $\alpha\in\Omega$ and $x\in G$. Distributivity is then expressed by the identity $(xy)^\alpha=x^\alpha y^\alpha$.

In a group with operators $G$, each operator defines an endomorphism of the underlying group structure; these endomorphisms will sometimes be called the homotheties of the group with operators $G$.

A group with operators $G$ is called commutative (or Abelian) if its group law is commutative.

In what follows a group $G$ will be identified with the group with operators in $\varnothing$ obtained by giving $G$ the unique action of $\varnothing$ on $G$. This allows us to consider groups as special cases of groups with operators and to apply to them the definitions and results relating to the latter which we shall state.