Lagrange's theorem (group theory)
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 n variables has its variables permuted in all n ! ways, the number of different polynomials that are obtained is always a factor of n !. (For example, if the variables x, y, and z are permuted in all 6 possible ways in the polynomial x + y − z then we get a total of 3 different polynomials: x + y − z, x + z − y, and y + z − x. Note that 3 is a factor of 6.) The number of such polynomials is the index in the symmetric group S of the subgroup H of permutations that preserve the polynomial. (For the example of x + y − z, the subgroup H in S contains the identity and the transposition (x y).) So the size of H divides 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 (Z / p Z) ∗ {\displaystyle (\mathbb {Z} /p\mathbb {Z})^{*}}, the multiplicative group of nonzero integers modulo p, where p is a prime. In 1844, Augustin-Louis Cauchy proved Lagrange's theorem for the symmetric group S.

Camille Jordan finally proved Lagrange's theorem for the case of any permutation group in 1861.
combination:two_responses, detectable_content:number_placeholders[6], punctuation:no_comma