Chapter 3.2, Problem 26E

Suppose $G$ is a finite set with $n$ distinct elements given by $G=\left\{a_1,a_2,\mathrm{...},a_n\right\}$. Assume that $G$ is closed under an associative binary operation $\ast$ and that the following two cancellation laws hold for all $a,x,$ and $y$ in $G$:

$a\ast x=a\ast y$ implies $x=y;$

$x\ast a=y\ast a$ implies $x=y$.

Prove that $G$ is a group with respect to $\ast$.