Normal Subgroup Counterexample Im having trouble with the second part of this question,
Let $H$ be a normal subgroup of $G$ with $|G:H| = n$,
i) Prove $g^n \in H$ $\forall g \in G$ (which i have done)
ii) Give an example to show that this all false when $H$ is not normal in $G$.(which I am having trouble with showing)
Any suggestions?

Hint: If $H$ is not normal in $G$, then $G$ is necessarily nonabelian. Consider the smallest nonabelian group you know.
