Coset Decomposition

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

This partition is called right coset decomposition of G. The procedure to obtain distinct members of this partition is given below:
H itself is a right coset. Now suppose a \in G and a \notin H then Ha will be another distinct right coset. Again let b be another such element that b \in G and b \notin H and also b \notin Ha then Hb will be another distinct right coset. Proceeding in this way all distinct right cosets of H in G will be obtained.

Thus G = H \cup Ha \cup Hb \cup Hc \ldots where a,b,c are elements of G so chosen that all right cosets are distinct. In the same way left coset decomposition of G can be obtained.