Complement of a normal subgroup ($G=NH,N\cap H=1$)

A group complement to a normal subgroup $N◃G$ is a subgroup $H$ with $G=NH$ and $N\cap H=1$. Projection identifies $H$ with $G/N$, and conjugation gives $G=N⋊H$. This is a subgroup notion, not the set-theoretic complement of $N$.