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Suppose we have a non-empty set $P$ equipped with an associative binary operation $\bullet$ such that for every $a \in P$ there exists a unique $b \in P$ with $aba=a$. How would we go about proving this is a group?
I have tried various things, and proved some smaller results such as for the element $b$, the corresponding unique element $c$ such that $bcb=b$ satisfies $c=a$, but every attempt to show this structure is in fact a group seems to rely on circular logic that either a unique identity exists, or each element has a unique inverse, both of which we obviously have to prove!
Any help would be much appreciated.