Find order of $xy$ provided $x^2=e, y^3=e$ and $yxy=xy^2x$ Let $G$ be a group and $x,y \in G$.  Suppose $x^2=e$, $y^3=e$ and $yxy=xy^2x$. Find the order of $xy$.
I have no idea what to do. I am suspecting $(xy)^6=e$ (order 6) but I don't know how to solve it.

When in doubt, look at the successive powers of $xy$ and try to simplify.  Here, we have:
$$
xy\\
(xy)^2=xyxy=x(yxy)=x(xy^2 x)=x^2y^2x=y^2x\\
(xy)^3=xy(y^2x)=xy^3x=xx=e
$$
So, in fact, $(xy)^3=e$, which means that the order of $xy$ is (at most) $3$.
