History
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 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 Sn of the subgroup H of permutations that preserve the polynomial. (For the example of x + y − z, the subgroup H in S3 contains the identity and the transposition (xy).) 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.
Lagrange did not prove his theorem; all he did, essentially, was to discuss some special cases. The first complete proof of the theorem was provided by Abbati and published in 1803.
Read more about this topic: Lagrange's Theorem (group Theory)
Famous quotes containing the word history:
“In front of these sinister facts, the first lesson of history is the good of evil. Good is a good doctor, but Bad is sometimes a better.”
—Ralph Waldo Emerson (1803–1882)
“History is not what you thought. It is what you can remember. All other history defeats itself.
In Beverly Hills ... they don’t throw their garbage away. They make it into television shows.
Idealism is the despot of thought, just as politics is the despot of will.”
—Mikhail Bakunin (1814–1876)
“Look through the whole history of countries professing the Romish religion, and you will uniformly find the leaven of this besetting and accursed principle of action—that the end will sanction any means.”
—Samuel Taylor Coleridge (1772–1834)