Let $G$ a group. Let $x\in G$. Assume that for every $y\in G, xyx=y^3$. Prove that $x^2=e$ and $y^8=e$ for all $y\in G$. To put this in context, this is my first week abstract algebra.
Let $G$ a group. Let $x\in G$. Assume that for every $y\in G, xyx=y^3$.
Prove that $x^2=e$ and $y^8=e$ for all $y\in G$.
A hint would be appreciated.

Hint: The object $x$ is fixed, I would rather call it $a$. I assume you have proved that $a^2=e$.
For any object $u$, we have $aua=(u)(u)(u)$. Let $u=aya$.  
