Lagrange's Theorem

Lagrange’s Theorem. Let $G$ be a finite group , and let $H \le G$ be a subgroup . Then all left cosets of $H$ in $G$ have the same cardinality as $H$, the distinct left cosets form a partition of $G$, and

where $[G:H]$ is the index of $H$ in $G$. In particular, $|H|$ divides $|G|$.

This is the basic divisibility theorem for finite groups and is the starting point for many counting arguments. A standard consequence is the fact that the order of an element divides the order of the group .

Proof sketch. Pick one representative from each left coset; multiplying by that representative gives a bijection $H \to gH$, so each coset has size $|H|$. Since cosets are disjoint and cover $G$, the total size of $G$ is (number of cosets)$\times |H|$.