Remark

(1) If $A$ is any subset of a group $G$ and $H$ is a normal subgroup of $G$, then $AH = HA$; this set is obtained by saturating $A$ with respect to the relation $x \equiv y \pmod{H}$.

(2) If $H$ is a normal subgroup of $G$ of finite index, the quotient group $G/H$ is a finite group of order $(G:H)$.

Note that if $H$ is a normal subgroup of a group $G$ and $K$ is a normal subgroup of $H$, $K$ is not necessarily a normal subgroup of $G$ (I, $§ 5$, Exercise 10).

Let $G$ be a group with operators. The intersection of every family of normal stable subgroups of $G$ is a normal stable subgroup. Hence, for every subset $X$ of $G$, there exists a smallest normal stable subgroup containing $X$, called the normal stable subgroup generated by $X$.

In a group with operators $G$, the stable subgroups $G$ and $\{e\}$ are normal.