Every subgroup of a group induces an important decomposition of .

We see that the left and the right coset determined by the same element need not be equal. We may also notice some interesting regularities: all the cosets have equal sizes; some of them are equal (e.g. ), and the others are disjoint (e.g. ). These are not accidents.

We give one interesting application.

Edmund F Robertson

11 September 2006