Given a group and subgroup , the order of the subgroup divides the order of the group. #m/thm/group
This is often stated as
where is the number of unique (left or right) cosets of ,
and is called the Lagrange index.
Proof
Let .
Any element is contained at least in the coset.
Since Cosets are either identical or disjoint,
cosets form a Partition of .
Since is finite there is a finite number of cosets in the partition .
The number of elements in each coset is equal to .
Therefore, .
Corollary
The order of an element divides the order of a finite group ,
since forms a Cyclic subgroup.