Proving normality of group of elements to the power of $n$ and that the order of every element in the quotient group is finite.

Question

Let $G$ be a group so that for some $n \in \mathbb{N} $ $(ab) ^n=a^{n}b^{n}, \forall a, b \in G. $ Prove that $G^{n} =\left\{g^n, g\in G \right\} $ is a normal subgroup of $G$, and that the order of every element in the quotient group $G/G^n$ is finite.

Edit: I proved the normality, and for the other part the order of every coset $aG^n$ has to be less than or equal to $n$ because when we look at it to the power od $n$ we get $G^n$ so it is finite. Is this correct?

Answer 1

That seems like a good idea, and it is the way I would have proven it.