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Possible Duplicate:
Prove if an element of a monoid has an inverse, that inverse is unique

How to show that the left inverse x' is also a right inverse, i.e, x * x' = e?

Also, how can we show that the left identity element e is a right identity element also?

Thanks

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    Double checking the title for typos is usually a great idea!2012-02-02
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    (Presumably you are in a group or something? Add details to the body of the question so that it makes sense :) )2012-02-02
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    Same ground covered: http://math.stackexchange.com/questions/102882/prove-if-an-element-of-a-monoid-has-an-inverse-that-inverse-is-unique/102883#1028832012-02-02
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    It depends on the definition of a group that you are using. If you define a group to be a set with associative binary operation such that there exists a left identity $e$ such that all elements have left inverses with respect to $e$ then showing that left identity/inverses are unique and also right identity/inverses can be a challenging exercise.2012-02-02
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    @Derek Bingo-that was the point of my proof below and corresponding response to Dylan. When I first learned algebra, my professor DID in fact use those very weak axioms and go through this very tedious-but enlightening-process.2012-02-02

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