Prove that if for all $aba=bab$ then $|G|=1$.

Prove that if for all $aba=bab$ then $|G|=1$.

Let $G$ be a group such that for all $a,b \in G$ we have $aba=bab$. Prove
that $|G|=1$.
So I have to show that $G =\left\{e \right\} $. Because for any $a,b \in
G$ we have $aba=bab$, let $b=e$. Then $aea=eae$ so $a^2 = a$ hence $a =
e$. Because $a$ is any we have $G=\left\{e \right\} $.
Does it work?