Lagrange's Theorem - Imported Knowlpedia Content

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

|G| = [G:H]\cdot |H|,

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

Remarks

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.