Showing that $G^n$ is a normal subgroup of $G$

Question

This is from exercise 2.55 "An Introduction to the Theory of Groups" by Joseph J. Rotman.

Let $G$ be a finite group, such that for some fixed integer $n>1$,for all $x,y\in G$, $(xy)^n = x^ny^n$. I need to prove that $G^n=\{x^n:x\in G\}$ is a normal subgroup of $G$.

I could prove that $G^n$ is a subgroup but i can't figure out the normal part.

Answer 1

Let $x \in G^n$. For $h \in G$, consider $h^{-1}xh.$ Write $x=g^n$ for some $g$. Then $h^{-1} x h=h^{-1}g^nh =(h^{-1} g h)^n$ because all of the $h$ terms in the middle cancel. Hence, $h^{-1} x h\in G^n$, and so $G^n$ is normal.