Cauchy's Theorem (Finite Groups)

Cauchy’s Theorem (Finite Groups). Let $G$ be a finite group , and let $p$ be a prime number such that $p \mid |G|$. Then there exists an element $g \in G$ with $g \ne e$ and $g^p = e$; equivalently, $G$ has a cyclic subgroup of order $p$.

Cauchy’s theorem is a partial converse to Lagrange's theorem : instead of saying “subgroup orders divide $|G|$,” it guarantees the existence of elements of certain prime orders when that prime divides $|G|$. It is a key input for Sylow's first theorem .