I. Let $G$ be a group in which, for some integer $n\gt 1$, $(ab)^n=a^n b^n$ for all $a,b\in G$. Show that
$G^{(n)}=\{x^{n}|x\in G\}$ is a normal subgroup of $G$.
$G^{(n−1)}=\{x^{n−1}|x\in G\}$ is a normal subgroup of $G$.
II. Let $G$ be as in the problem above. Show that
$a^{n−1}b^n=b^n a^{n−1}$ for all $a,b\in G$.
$(aba^{−1}b^{−1})^{n(n−1)}=e$ for all $a,b\in G$.