Let be a subgroup of group . We know that no right coset of in is empty and any two right cosets of in are either disjoint or identical. The union of all right cosets of in is equal to . Hence the set of all right cosets of in gives a partition of .

This partition is called right coset decomposition of . The procedure to obtain distinct members of this partition is given below:

itself is a right coset. Now suppose and then will be another distinct right coset. Again let be another such element that and and also then will be another distinct right coset. Proceeding in this way all distinct right cosets of in will be obtained.

Thus where are elements of so chosen that all right cosets are distinct. In the same way left coset decomposition of can be obtained.