Cauchy's theorem is the first major existence theorem in finite group theory. The focus keyword is Cauchy's theorem, because the theorem says that whenever a prime $p$ divides the order of a finite group $G$, the group must contain an element of order $p$. Lagrange's theorem tells us that the order of a subgroup divides $|G|$; Cauchy's theorem gives a partial converse for prime divisors. This result begins the route toward p-groups, p-subgroups, and the Sylow theorems, where prime-power divisors control the internal structure of finite groups.

Let $G$ be a group and let $p$ be a prime number. An element $a\in G$ is said to have \textbf{prime order} $p$ if $a^p=e$ and $a^k\neq e$ for every integer $k$ satisfying $1\leq k<p$.