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Let $(S,*)$ be a semigroup such that for every $a \in S$ there exists $b \in S$ such that $a*b*a=a$. Prove that $S$ is a group. I tried first to prove that $1$ exists but I can't get my head around it. Any suggestions?

user53970
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Such a semigroup is called regular, and need not be a group in general. For a counterexample see here. However, if we require that for each $a\in S$ there is a unique $b\in S$ with $aba=a$, then $S$ is a group - this is a duplicate, see here.

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Dietrich Burde
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