# normal closure

Let $S$ be a subset of a group $G$. The normal closure of $S$ in $G$ is the intersection of all normal subgroups of $G$ that contain $S$, that is

 $\bigcap_{S\subseteq N\trianglelefteq G}\!\!N.$

The normal closure of $S$ is the smallest normal subgroup of $G$ that contains $S$, and so is also called the normal subgroup generated by $S$.

It is not difficult to show that the normal closure of $S$ is the subgroup generated by all the conjugates of elements of $S$.

The normal closure of $S$ in $G$ is variously denoted by $\langle S^{G}\rangle$ or $\langle S\rangle^{G}$ or $S^{G}$.

If $H$ is a subgroup of $G$, and $H$ is of finite index in its normal closure, then $H$ is said to be nearly normal.

 Title normal closure Canonical name NormalClosure1 Date of creation 2013-03-22 14:41:50 Last modified on 2013-03-22 14:41:50 Owner yark (2760) Last modified by yark (2760) Numerical id 9 Author yark (2760) Entry type Definition Classification msc 20A05 Synonym normal subgroup generated by Synonym conjugate closure Synonym smallest normal subgroup containing Related topic Normalizer Related topic CoreOfASubgroup Defines nearly normal