Problem in Group theory about normal subgroup

Question

Let $H$ be a normal subgroup of a group $G$ with index $n$. Prove that if $g \in G$ then $g^n\in H$. Find an example pointing that this can not occur if $H$ is not normal.

Answer 1

In general, for any group $G$ with $n=|G|$, $g^n=e$ where $e$ is the identity element. So, if $H$ is a normal subgroup of $G$ with $[G:H]=n$, then $|G/H|=n$. Let $g\in G$, then $H=eH=(gH)^n=g^nH$. Hence, $g^n\in H$.