Lagrange did not prove Lagrange's 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: . 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 of abstract groups, this result of Lagrange on polynomials was recognized to extend to the general theorem about finite groups which now bears his name.