**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**

*|G|=|H|.[G:H]*

**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>*.