Cauchy's Lemma (Group Theory)

Theorem
Let $\struct {G, \circ}$ be a group of finite order whose identity is $e$.

Let $p$ be a prime number which divides the order of $G$.

Then $\struct {G, \circ}$ has an element of order $p$.

Also see

 * Cauchy's Group Theorem, which establishes that, given the same conditions, $G$ also has a subgroup of order $p$.