Also, a subgroup, H, of a group, G, is a normal subgroup if and only if the "left cosets" and "right cosets" are the same. A result of that is that we can define the group operation on the cosets (if p is in coset A and q is in coset B then AB is the coset that contains pq) in such away that the collection of cosets is a group in its own right: G/H.