IMC 2005 · Problem 6

Given a group $G$, denote by $G(m)$ the subgroup generated by the $m$th powers of elements of $G$. If $G(m)$ and $G(n)$ are commutative, prove that $G(\gcd(m,n))$ is also commutative. ($\gcd(m,n)$ denotes the greatest common divisor of $m$ and $n$.)