Let $G$ be a finite set and $*$ a binary operation on $G$ such that:
- The operation $*$ is associative.
- For all $x$, $y$, and $z$ in $G$, if $x*y=x*z$ then $y=z$ and if $y*x=z*x$ then $y=z$.
I must show that this set has an identity element $e$ in $G$ such that for all $x$ in $G$, $e*x=x$. I'm not sure how to start at all. I was thinking of showing that for some arbitrary element in $G$, there exists e such that $e*a=a$ and then proving it for the rest of the elements.