Complement (reviewed)

If $H \trianglelefteq G$ is a normal subgroup, a complement of $H$ is a subgroup $K \subseteq G$ with $\lvert H \cap K \rvert = 1$ and $HK=G$ where $$HK = \{ hk \mid h\in H \text{ and } k\in K\}.$$

In order for a complement to exist, the sequence $$1 \to H \to G \to G/H \to 1$$ must split, in which case the complements are precisely the images of the possible splittings $G/H \to G$. All complements are isomorphic to $G/H$, and if $K$ is any complement then $G$ can be described as the internal semidirect product $H \rtimes K$.