Lagrange's theorem

Lagrange's theorem is a result in group theory that relates the size (order) of a finite group to the size of its subgroups. It says that the number of elements in any subgroup of a finite group divides the number of elements in the whole group.

Important properties

• If $G$ is a finite group and $H$ is a subgroup of $G$, then $|H|$ divides $|G|$.

• The number of distinct left (or right) cosets of $H$ in $G$ is $|G|/|H|$.

• The order of any element in $G$ divides $|G|$.