TITLE: Suppose $H \le G$ and $g^2\in H$ for all $g\in G$. Show $H$ is a normal subgroup of $G$
QUESTION [9 upvotes]: Let $G$ be a group and $H$ a subgroup of $G$. Suppose $g^2\in H$ for all $g\in G$. Show $H$ is a normal subgroup of $G$. I tried lots of methods, but failed. Any suggestion? Thanks.

REPLY [11 votes]: Let $g \in G$, $h \in H$. We know that $g^2 \in H$. Hence $g^2 h \in H$. We have:
$$
(ghg^{-1})(g^2 h) = ghgh = (gh)^2 \in H
$$
Thus, $ghg^{-1} \in H$ and $H$ is a normal subgroup of $G$.