Hello!

If M is a finite monoid and in M implies that , show that is a unit.

with hints: If show that are distinct.

Attempt:

If then by definition it has an identity element . Then, because M is finite, there will be an element which will map to the identity of M. Each will be distinct as for every , u will map it back to itself. Thus, which, because will map to itself and so must be a unit.

Thanks guys!