Coset

Consider a countable Subgroup with Elements and an element not in , then

 (1)

 (2)

for , 2, ... are left and right cosets of the Subgroup with respect to . The coset of a Subgroup has the same number of Elements as the Subgroup. The Order of any Subgroup is a divisor of the Order of the Group. The original Group can be represented by
 (3)

For a not necessarily Finite Group with a Subgroup of , define an Equivalence Relation if for some in . Then the Equivalence Classes are the left (or right, depending on convention) cosets of in , namely the sets

 (4)

where is an element of .