Is a finite monoid with left cancellation property always a group?

Question

I need to answer and show if a Monoid with left cancellation property always a group.

I managed to show that it is correct when cancellation property holds for both left and right (that was part a of the question), but I have a feeling i proved it wrong because i didn't use the right side at all. what I did is :

M is finite though if $x \in M$ there is some $i \in N$ such that

$x = x^i$

then

$x*e = x*x^{i-1}$

so

$x^{i-1} = e$

then $x^{i-2}$ is the inverse of x

so i proved that G is a Group without using the right cancelation property.

where am I wrong..

any help will be appreciated

Answer 1

Yes, every finite monoid $M$ with left (or right) cancellation is a group, because the mapping $x\mapsto ax$ is injective, and since $M$ is finite, it is already bijective by the pigeonhole (box) principle. Hence $M$ is a group.
Your argument above is not correct as demonstrated in the comments, but it can be corrected by using the cancellation property.