TITLE: If order of the element $a^5$ is 12 can we make any guess about the order of the element $a$ in a group $G$?
QUESTION [7 upvotes]: If order of the element $a^5$ is 12 can we make any guess about the order of the element $a$ in a group $G$?
Could anybody clear my this doubt?
Thanks for the help

REPLY [7 votes]: Hint: Let order of element $a$ is $n$
then $order (a^m) = \dfrac {n}{(m, n)}$, where $(m, n)$ is the gcd of $m$ and $n$.