Prove that if $H$ is a normal subgroup of a group $G$ and has index $n$, then for any $g\in G$ we have $g^n\in H$.
Give an example for if $H$ is not normal, the mentioned statement is not correct.
(Please give an example, except the symmetric group :D )
Hint: $x \in H \iff xH =H$. $xH$ is an element of the group $G/H$- a group because $H$ is normal. It follows that $H = e_{G/H} = (xH)^{[G:H]}=x^{[G:H]}H$. Look for a non-abelian group of small order for your counterexample.