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>.