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Do left sided identity and right sided inverse suffice to recover both sided axioms in group theory?

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August 18, 2007 4:31AM

No. For a counterexample, define a*b=b for all a and b in the group. Then we can pick any e to be the left identity of all the elements. Similarly, any b has the right inverse e because b*e=e. However, (if there is more than one element), this doesn't satisfy the conditions on a group because there is no single (two-sided) identity element. If a*x=a and b*x=b, then x=a and x=b, which obviously can't hold in the general case.