Question on abstract algebra about invertiblity?

Question

Let $(G,*)$ be a group.

Could anyone give me an example $a,b\in G$ such that
$$a*b=e\mbox{ and } b*a\neq e $$ Where $e$ is the identity element.

I would appreciate any help. Thanks in advance!

Answer 1

This isn't possible in a group. If $ab = e$, then $b=a^{-1}$ and $a=b^{-1}$, which implies $ba=e$.