MathNet 2018-part-04 — 181 problems (EN–VI) Question 156

Let $p$ be a natural number greater than or equal to $2$ and let $(M, \cdot)$ be a finite monoid such that $a^{p} \neq a$ for every $a \in M \backslash\{e\}$, where $e$ is the identity element of $M$. Prove that $(M, \cdot)$ is a group.