Groups of prime order are cyclic

Proposition (Prime order implies cyclic). Let $G$ be a finite group with $|G|=p$ where $p$ is prime. Then $G$ is cyclic; more precisely, for every $g\in G$ with $g\neq e$, one has $G=\langle g\rangle$.

Context. This is one of the first applications of Lagrange's theorem : subgroup orders must divide the group order.