Let $G$ be a finite semi-group with identity such that it has only one idempotent.Is $G$ a group?
It only remains to show that for any $a\in G$ $\exists b\in G$ such that $ab=ba=e$ where $e$ is the identity of $G$
Also $e$ is the only idempotent of $G$ .How to proceed next?
Show that for every $x\in G$, there is an $n\in \Bbb N$ such that $x^n$ is idempotent. Then you can claim that for every $x\in G$, some power of $x$ equals $e$.