that Fix in , and let denote the double stabilizer }. Then the double stabilizer is a subgroup of . Because is a group, for each in there is precisely one in such that , namely ; however, may not be in . Similarly, for each in there is precisely one in such that , but may not be in . The double stabilizer therefore has the descriptions (Orbit–stabilizer theorem) There is a bijection between and under which corresponds to . It follows that if , , and are finite, then (Cauchy–Frobenius lemma) Let denote the elements fixed by the action of . Then In particular, if , , and are finite, then the number of double cosets equals the average number of points fixed per pair of group elements. There is an equivalent description of double cosets in terms of single cosets. Let and both act by right multiplication on . Then acts by left multiplication on the product of coset spaces . The orbits of this action are in one-to-one correspondence with . This correspondence identifies with the double coset . Briefly, this is because every -orbit admits representatives of the form , and the representative is determined only up to left multiplication by an element of . Similarly, acts by right multiplication on , and the orbits of this action are in one-to-one correspondence with the double cosets . Conceptually, this identifies the double coset space with the space of relative configurations of an -coset and a -coset. Additionally, this construction generalizes to the case of any number of subgroups. Given subgroups , the space of -multicosets is the set of -orbits of . The analog of Lagrange's theorem for double cosets is false. This means that the size of a double coset need not divide the order of . For example, let be the symmetric group on three letters, and let and be the cyclic subgroups generated by the transpositions and , respectively. If denotes the identity permutation, then This has four elements, and four does not divide six, the order of . It is also false that different double cosets have the same size. Continuing the same example, which has two elements, not four. However, suppose that is normal. As noted earlier, in this case the double coset space equals the left coset space . Similarly, if is normal, then is the right coset space