Normal complement

Definition with symbols
Suppose $$H$$ is a subgroup of a group $$G$$. A normal complement $$N$$ to $$H$$ in $$G$$ is a defining ingredient::normal subgroup of $$G$$ that is a defining ingredient::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 defining ingredient::retract.