Let be a group, any subgroup of , then the number of left (right) cosets of in is called the index of in , denoted as .
Lagrange's Theorem:
is a finite group, any subgroup of , then .
Proof: Let , then there are distinct left cosets of , denote , then is a partition of (Partition is not covering, cosets in are not overlapped), at this time we have
If right cosets are used, the same result is obtained.
According to the inference of Lagrange's theorem, the order of the elements in the group must divide 4, so it can only be 1, 2, or 4.
(1) If there is an element of order 4, then this group is a cyclic group, .
(2) If there is no element of order 4, then except for the identity element, the other three elements must have order 2. Let , is the identity, . Thus it is an abelian group (otherwise there exist , with , leading to , but , , hence , a contradiction). Since or or (otherwise or , or ), so . Similarly, , , thus it is the Klein four-group.
This example shows that there are only two groups of order 4 up to isomorphism: the cyclic group of order 4 or the Klein four-group.