By the same token, how do we prove a^{2} = b in the first place? $\noindent Well the group is $\{e, a, b\}$ (all \underline{distinct} elements). By closure of the group, $a^2$ must be one of $e, a$, or $b$. Now, what would happen if $a^{2} = e$ or if $a^{2} = a$?$