# Cauchy's Theorem (Group theory) Let $G$ be a finite group s.t. $p\mid |G|$ then $G$ has an element of order $p$. It is a partial converse of Lagrange Theorem as it state that if there $p$ divide $|G|$ then there is a subgroup of size $p$. It is not "full" converse because it is limited to prime divisor and not general divisors. Proof: define a set $X:=\{(g_1,...,g_p)\in \prod_{i=0}^p G : g_1\cdots g_p=e\}$ and action of the generator of $C_p$ as $(g_1,...,g_p)\mapsto (g_2,...,g_p,g_1)$. This is a valid action as $(g_2,...,g_p,g_1)$ satisfied $g_2\cdots g_p\cdot g_1=g_1^{-1}(g_1g_2\cdots g_p)\cdot g_1=1$ . a fixed orbit satisfies $g_1=g_2=\cdots=g_p$ i.e., it is of the form $(g,...,g)$ s.t. $g^p=e$. note that we need to show that $g\ne e$ to prove the theory. As every element in $X$ is determined by $p-1$ elements (and the last element is the inverse of the multiplication of previous elements) - we have $|X|=|G|^{p-1}$ i.e, $p \mid |X|$. Also, as all non-fixed orbits are of size $p$ then the number of fixed orbit must also be divisible by $p$ and so there is more than one fixed orbit i.e., there is a fixed orbit of the form $(g,...,g)$ s.t., $g\ne e$ as needed. ## Created 2026-05-29 11:32