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