# Proving this non-empty set and binary operation is a group [duplicate]

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.

## marked as duplicate by Theo Bendit, nmasanta, Bill Dubuque, José Carlos Santos, Daniele TampieriSep 12 at 6:44

For example, let $$P$$ consist of any two distinct reflections of a regular polygon. Then, under the usual composition of symmetries, all the required conditions are satisfied but $$P$$ is not a group.