What does $x^n=y^n$ imply for $x=\pm y$ in a group? I am wondering if the following statement is true:


  If $G$ is a group, then for all $x,y \in G, x^n=y^n$ implies $x=y$ or $x=-y$.


I have tried using an induction argument and it might work by using casework on whether $n$ is odd or even, but I'm not sure.



EDIT: If $G$ is a group, then if $x,y \in G$ and $ x^n=y^n$, then what information does this give us about $x$ and $y$?
No. If the group has order $n$ then $x^n = e$ (the group identity) for every $x$.
If $x^n = y^n$, then $x^n y^{-n} = 1$, and hence $(xy^{-1})^n = 1$. In other words, $x$ times the inverse of $y$ is idempotent.