Normal subgroup is permutable!

Question

Where can I find the proof of : The normal subgroups are always permutable?? The group is permutable if $HX=XH$ for every subgroup $X$ of group $G$( $H$ is a subgroup of the group $G$).

Answer 1

Let $H$ be a normal subgroup of $G$. Then $xH=Hx$ for any element $x\in G$, so $XH=HX$ for any subset $X$ of $G$ (and the subgroups $X$ of $G$ in particular).

If by "normal subsemigroup" we mean that $N$ is a subsemigroup of (semigroup) $G$ such that:

$$ Ng = gN \; \forall g\in G $$

then the same sort of proof shows $NX = XN$ for any subset $X$ of $G$.