Let $\Sigma$ be a smooth oriented surface (possibly with boundary) and $G:=\pi_1(\Sigma)$ its fundamental group. Suppose that two nonzero elements $\alpha$, $\beta\in G$ satisfies $\alpha \beta = \beta^m \alpha^n$ holds for some $m,n\in\mathbb{Z}$. Then my claim is that we must have $m=n =1$.
By abelinization, we have $\alpha^{n-1} = \beta^{1-m}\in H^1(\Sigma)$ but I could not proceed more. I'll appreciate if someone suggests an idea or more general statements on fundamental groups of surfaces.