Normal complement

Definition

Definition with symbols

Suppose $H$ is a subgroup of a group $G$. A normal complement $N$ to $H$ in $G$ is a normal subgroup of $G$ that is a permutable complement to $H$ in $G$. In other words, $N\cap H$ is trivial and $NH=G$.

Every subgroup need not have a normal complement. A subgroup that possesses a normal complement is termed a retract.