Editing Lagrange's theorem (group theory) (section)

== History ==
Lagrange himself did not prove the theorem in its general form. He stated, in his article ''Réflexions sur la résolution algébrique des équations'',<ref>{{citation | last = Lagrange|first= Joseph-Louis | author-link= Joseph-Louis Lagrange | year = 1771 | title = Suite des réflexions sur la résolution algébrique des équations. Section troisieme. De la résolution des équations du cinquieme degré & des degrés ultérieurs. |trans-title=Series of reflections on the algebraic solution of equations. Third section. On the solution of equations of the fifth degree & higher degrees | journal = Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Berlin | pages = 138–254 | url = https://books.google.com/books?id=_-U_AAAAYAAJ&pg=PA138}} ; see especially [https://books.google.com/books?id=_-U_AAAAYAAJ&pg=PA202 pages 202-203.]</ref> that if a polynomial in {{mvar|n}} variables has its variables permuted in all {{math|''n''!}} ways, the number of different polynomials that are obtained is always a factor of {{math|''n''!}}.  (For example, if the variables {{mvar|x}}, {{mvar|y}}, and {{mvar|z}} are permuted in all 6 possible ways in the polynomial {{math|''x'' + ''y'' − ''z''}} then we get a total of 3 different polynomials: {{math|''x'' + ''y'' − ''z''}}, {{math|''x'' + ''z'' − ''y''}}, and {{math|''y'' + ''z'' − ''x''}}.  Note that 3 is a factor of 6.) The number of such polynomials is the index in the [[symmetric group]] {{math|''S''<sub>n</sub>}} of the subgroup {{mvar|''H''}} of permutations that preserve the polynomial.  (For the example of {{math|''x'' + ''y'' − ''z''}}, the subgroup {{mvar|''H''}} in {{math|''S''<sub>3</sub>}} contains the identity and the transposition {{math|(''x y'')}}.) So the size of {{mvar|''H''}} divides {{math|''n''!}}.   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.

In his ''[[Disquisitiones Arithmeticae]]'' in 1801, [[Carl Friedrich Gauss]] proved Lagrange's theorem for the special case of <math>(\mathbb Z/p \mathbb Z)^*</math>, the multiplicative group of nonzero integers [[Modular arithmetic|modulo]] {{mvar|p}}, where {{mvar|p}} is a prime.<ref>{{Citation|last=Gauss|first=Carl Friedrich|author-link=Carl Friedrich Gauss|title=Disquisitiones Arithmeticae|location=Leipzig (Lipsia)|language=la|publisher=G. Fleischer|year=1801}}, [http://babel.hathitrust.org/cgi/pt?id=nyp.33433070725894;view=1up;seq=63 pp. 41-45, Art. 45-49.]</ref>  In 1844, [[Augustin-Louis Cauchy]] proved Lagrange's theorem for the symmetric group {{math|''S''<sub>n</sub>}}.<ref>[[Augustin-Louis Cauchy]], ''§VI. — Sur les dérivées d'une ou de plusieurs substitutions, et sur les systèmes de substitutions conjuguées'' [On the products of one or several permutations, and on systems of conjugate permutations] of:  ''"Mémoire sur les arrangements que l'on peut former avec des lettres données, et sur les permutations ou substitutions à l'aide desquelles on passe d'un arrangement à un autre"'' [Memoir on the arrangements that one can form with given letters, and on the permutations or substitutions by means of which one passes from one arrangement to another] in:  ''Exercises d'analyse et de physique mathématique'' [Exercises in analysis and mathematical physics], vol. 3 (Paris, France:  Bachelier, 1844), [https://books.google.com/books?id=-c3fxufDQVEC&pg=PA183 pp. 183-185.]</ref>

[[Camille Jordan]] finally proved Lagrange's theorem for the case of any [[permutation group]] in 1861.<ref>{{citation | first=Camille| last=Jordan|author-link=Camille Jordan | year = 1861 | title = Mémoire sur le numbre des valeurs des fonctions |trans-title=Memoir on the number of values of functions |journal = Journal de l'École Polytechnique | volume = 22 | pages = 113–194 | url = http://gallica.bnf.fr/ark:/12148/bpt6k433691p/f118.image.langEN}}  Jordan's generalization of Lagrange's theorem appears on [http://gallica.bnf.fr/ark:/12148/bpt6k433691p/f171.image.r=Lagrange.langEN page 166.]</ref>