the remaining 5 elements of must come from distinct pairs of elements in that are not in . This is impossible since pairs of elements must be even and cannot total up to 5 elements. Thus, the assumptions that is wrong, so . Then, where , must be in the form where are distinct elements of . The other four elements in are cycles of length 3. Note that the cosets generated by a subgroup of a group form a partition of the group. The cosets generated by a specific subgroup are either identical to each other or disjoint. The index of a subgroup in a group is the number of cosets generated by that subgroup. Since and , will generate two left cosets, one that is equal to and another, , that is of length 6 and includes all the elements in not in . Since there are only 2 distinct cosets generated by , then must be normal. Because of that, . In particular, this is true for . Since . Without loss of generality, assume that , , , . Then , , , , . Transforming back, we get . Because contains all disjoint transpositions in , . Hence, . Since , we have demonstrated that there is a third element in . But earlier we assumed that , so we have a contradiction. Therefore, our original assumption that there is a subgroup of order 6 is not true and consequently there is no subgroup of order 6 in and the converse of Lagrange's theorem is not necessarily true. Q.E.D. History Lagrange himself did not prove the theorem in its general form. He stated, in his article Réflexions sur la résolution algébrique des équations, that if a polynomial in variables has its variables permuted in all ways, the number of different polynomials that are obtained is always a factor of . (For example, if the variables , , and are permuted in all 6 possible ways in the polynomial then we get a total of 3 different polynomials: , , and . Note that 3 is a factor of 6.) The number of such polynomials is the index in the symmetric group of the subgroup of permutations that preserve the polynomial. (For the example of , the subgroup in contains the identity and the transposition .) So the size of divides . With the later development