Is there a simple proof of the following fact:


**Fact**. Let $S$ be a completely simple semigroup with cancellations, i.e. each of the equalities $xa=xb$, $ax=bx$ implies $a=b$. Prove that $S$ is a group.


Using Sushkevich-Rees Theorem, I can prove it, but my proof is not elegant.


Can you prove this fact using only simple arguments? Probably, you know a paper, where this result was proved at the first time?



A completely simple semigroup with cancelation is a group (simple proof)
A completely simple semigroup has by definition a (primitive) idempotent $e$ and $SxS=S$ for all $x\in S$. If $x\in S$, then $eex=ex$ and so $ex=x$ and similarly, we see that $e$ is a right identity. Thus $S$ is a monoid and the identity is the unique idempotent of $S$. Let us write $1$ for the identity. Let $x\in S$. Then $uxy=1$ for some $u,y$ by simplicity. Then $xyuxyu=xyu$ and so $xyu=1$ as $1$ is the unique idempotent. Similarly, $yuxyux=yux$ and hence $yux=1$ because each idempotent is the identity. Thus $x$ is invertible with inverse $yu$. So $S$ is a group.


