Are all finite monoids groups?
If I have a monoid $M$ such that $|M|=c$ for an integer $c$, then for all $x\in M$, we should have $x^k=e$ for some minimal $k$. Then, we have $x^{-1}=x^{k-1}$.
I don't see anything wrong with my proof, but I haven't found an answer online, and it doesn't feel like a `trivial' result, so I'm wondering if this is true.
You argument is definitely wrong, but it can be modified to get a property related to groups. Take an element $x$ of a finite monoid $M$. First show that $x$ has an idempotent power, that is, for some $k > 0$, $x^k = x^{2k}$ (or see here for a proof). Next show that there is a maximal subsemigroup of $M$ which is a group with $x^k$ as identity.