Let $G$ be a group in which, for some integer $n > 1$, $\ (ab)^n = a^n b^n$ for all $a$, $b \in G$. Let $G^{(n-1)} \colon = \{ x^{n-1} \ | \ x \in G \}$. Is $G^{(n-1)}$ a subgroup of $G$? I know that, once we have demonstrated it to be a subgroup, then it is a normal subgroup.
Furthermore, I also know that the subset $G^{(n)} \colon= \{x^n \ | \ x \in G \}$ is a subgroup (rather a normal subgroup) of $G$.