Let G be a finite group, and let S and T be 2 subsets of G such that G does not equal ST. Show that
There are |S| elements in S. There are |T| elements in , and also in its coset . Also, the sets S and are disjoint: for suppose that an element is also in . Then for some . But that implies that , which contradicts the choice of g. Therefore there are |S|+|T| distinct elements in the set , and so .