When is $G^n$ a subgroup of $G$?

Question

Let $G$ be a group and $n > 1$ be a positive integer. I want to know when $G^n = \{ g^n | g \in G \}$ is a subgroup of $G$? A general answer would be most helpful, though in the context I am using this, the most useful things we can say is that $G$ is nilpotent and $n$ is prime, I am not sure if that is useful.

Answer 1

This is true when $G$ is a regular $p$-group and $n$ is a power of $p$.

Answer 2

Every divisible group has this property: $G^n=G$ in this case. Examples of divisible groups exist among abelian groups, such as ${\mathbb Q}$, and even finitely generated infinite nonabelian groups...