Theorem Let G be a finite group and let H be a subgroup. Then the order of H divides the order of G.

The proof of this theorem comes down to the fact that the cosets of H in G partition G. In fact if we write [G:H] for the number of such cosets (also called the index of H in G) we have

The Counting Formula


Corollary The order of an element of a finite group divides the order of the group.

The proof follows from the fact that if a is an element of G then the order of a is the same as the order of the cyclic subgroup it generates <a>.