I am asked to show that if E is a semi-group and if
(i) there is a left identity in E
(ii) there is a left inverse to every element of E
then, E is a group.
The Attempt at a Solution
Well I can't seem to find the solution, but it's very easy if one of the two "left" above is replaced by a "right". For instance, if we replace the existence of a left inverse condition by the existence of a right inverse, then we find that the left identity is also a right identity like so:
Let a,b be in E. Then ab=a(eb)=(ae)b ==> a=ae (by multiplying by bˉ¹ from the right). So e is a right identity also. Then it follows that every right inverse is also a left inverse:
aaˉ¹=e ==>(aaˉ¹)a=ea ==>a(aˉ¹a)=a ==> (aˉ¹a)=e.
So, does anyone know for a fact that this question contains or does not contain a typo?