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?

See my answers here If $[G:H]=n$, is it true that $x^n\in H$ for all $x\in G$?. You can take $H$ to be any non-normal Sylow $p$-subgroup of $G$.
