Corollary

*Let $G$ be a group and $H$ and $K$ two subgroups of $G$ such that $H$ is normal, $H \cap K = \{e\}$ and $H.K = G$. Let $\tau$ be the operation of $K$ on $H$ by inner automorphisms of $G$. The mapping $(h, k) \mapsto hk$ is an isomorphism of $H \times_{\tau} K$ onto $G$.

Under the hypotheses of this corollary, $G$ is said to be the semi-direct product of $K$ by $H$.