Algebra problem solved 2

Problem: "Prove that a finite monoid in which the cancellation law holds is a group. "

To prove it, we must show that for all a there is an inverse element.

We know that the monoid in question is finite so:

$\forall a \exists n$ such that aⁿ = e where e is the identity element.

This stems from the fact that monoids are closed under their operation.

aⁿ= e implies a^n-1a= e. So the inverse element is a^n-1

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