Crossed Modules

A concept arising from the concept of a $G$-module (see Module). A group $M$ (not necessarily Abelian) with a group of operators $G$ and homomorphism $f : M \rightarrow G$ such that for any $g \in G$ and all $x,y \in M$, $$ f(gx) = g f(x) g^{-1}, \ \ \ f(x)y = x y x^{-1} \ . $$ is called a crossed $(G,f)$-module. $M$ is a $G$-module (i.e. its underlying group is Abelian) if and only if $f$ is constant, equal to $e \in G$.