Example

In a commutative group $G$, written multiplicatively, $(xy)^n=x^ny^n$ for all $n\in\mathbf{Z}$ (§ 2, no. 8, equation (1)); the action $n\mapsto(x\mapsto x^n)$ of $\mathbf{Z}$ on $G$ therefore defines, together with the group law, the structure of a group with operators on $G$.