Let G be a group, and let $a,b \in G$. The order of an element $g \in G$ is defined as the smallest positive integer $n$ such that $g^n=e$, where $e$ is the identity element of $G$.
I want to prove that in any group, the elements $ab$ and $ba$ always have the same order. So far, I have noticed that if $ab$ and $ba$ commute, then the claim follows trivially. However, I suspect this should hold more generally for arbitrary elements $a$ and $b$.
I would appreciate a clear and rigorous proof for this result. Thank you!
