TITLE: For any $n$, $G^\dot{n}$ is a subgroup of $G$. Is $G$ abelian?
QUESTION [7 upvotes]: Let $G$ be a group such that for each $n$:
$$G^\dot{n}=\{a^n|a\in G\}$$
is a subgroup of $G$. Is $G$ abelian?

REPLY [4 votes]: Here it is a large family of those groups. A a finite  $p$-group is said to be powerful if $p$ is odd and $[G,G]\leq G^p$ or if $p=2$ and $[G,G]\leq G^4$. If $G$ is a powerful $p$-group then for any integer $n$ we have $G^n=\{g^n:n\in G\}$. This is Theorem 2.7 in the book Analytic pro-$p$ group: http://books.google.com.ar/books?id=7yxplSjMWtkC&printsec=frontcover&hl=es&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false