Closely related subgroups of a finite group

Let $G$ be a finite group and $n > 1$ an integer such that $(a b)^{n} = a^{n} b^{n}$ for all $a, b \in G$ . Let

G_{n} = {c \in G ∣ c^{n} = e} and G^{n} = {c^{n} ∣ c \in G}

You may take for granted that these are subgroups. Prove that both $G_{n}$ and $G^{n}$ are normal in $G$ , and $| G^{n} | = [G : G_{n}]$ .

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Let $G$ be a finite group and $n>1$ an integer such that $(ab)^n=a^n b^n$ for all $a,b\in G$. Let
 \[
	 G_n=\{c\in G\mid c^n=e\}\qquad\text{and}\qquad G^n=\{c^n\mid c\in G\}
\]
You may take for granted that these are subgroups. Prove that both $G_n$ and $G^n$ are normal in $G$, and $|G^n|=[G:G_n]$.