Free groups: $u^mv^k = v^ku^m \implies uv = vu$

Question

I'm trying to show that if $F_n$ is a free group of rank $n$ and $u,v \in F_n$ and $m, k >0$ then $u^mv^k = v^ku^m \implies uv = vu$. I can't seem to do it by manipulating the equation $u^mv^k = v^ku^m$ - are there any facts I'm missing? Thanks!

Answer 1

By definition, two words in a free group are equal if and only this follows by the group axioms, i.e., like $uv=uss^{-1}v$. So different reduced words are different in $F_n$. Hence the words $u^mv^k$ and $v^ku^m$ are only equal for some $m,k\ge 1$ if $u=v$.