Hello!

If M is a finite monoid and $\displaystyle au=bu$ in M implies that $\displaystyle a=b$, show that $\displaystyle u$ is a unit.

with hints: If $\displaystyle M=\{a_1,\cdots a_n\}$ show that $\displaystyle a_1u,\cdots , a_nu$ are distinct.

Attempt:

If $\displaystyle M=\{a_1,\cdots a_n\}$ then by definition it has an identity element $\displaystyle e$. Then, because M is finite, there will be an element which will map to the identity of M. Each $\displaystyle a_1u\cdots a_nu$ will be distinct as for every $\displaystyle a_n$, u will map it back to itself. Thus, $\displaystyle au=bu$ which, because $\displaystyle u$ will map $\displaystyle a$ to itself $\displaystyle a(e)=b(e)$ and $\displaystyle a=b(e)$ so $\displaystyle u$ must be a unit.

Thanks guys!