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Suppose we have a non-empty set $X$ with an associative binary operation, so that for all $x \in X$ there exists a unique $x'$ such that $xx'x=x$. Prove the set under this operation is a group.

Of course, associativity is already given, and unique inverses are obvious, but I have no idea how to go about showing an identity element exists, nor how to show the set is closed with respect to the operation. How can one go about this?

FireGarden
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    The set is closed with respect to the operation automatically the same way the associativity is already given. That's what “associative binary operation on $X$” means. Unique inverses are not obvious since you even cannot talk about inverses when you don't have identity element. – Adam Bartoš Jul 27 '13 at 11:04

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