Prove a semigroup is a group

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Homework Statement


Prove, that a senigruop S is a group if and only if for every x in S there exists a unique x' in S so that xx'x=x.


Homework Equations





The Attempt at a Solution


I am trying to prove that xx'=1S for all x in S. But all I can show is that xx' is in ES={x | x^2=x} for every x in S. I am having a lot of troubles proving xx'=yy' and I can't even prove that 1S si in S. Am I going in the right direction?
 

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