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:
“[Men say:] “Don’t you know that we are your natural protectors?” But what is a woman afraid of on a lonely road after dark? The bears and wolves are all gone; there is nothing to be afraid of now but our natural protectors.”
—Frances A. Griffin, U.S. suffragist. As quoted in History of Woman Suffrage, vol. 4, ch. 19, by Susan B. Anthony and Ida Husted Harper (1902)
“Three million of such stones would be needed before the work was done. Three million stones of an average weight of 5,000 pounds, every stone cut precisely to fit into its destined place in the great pyramid. From the quarries they pulled the stones across the desert to the banks of the Nile. Never in the history of the world had so great a task been performed. Their faith gave them strength, and their joy gave them song.”
—William Faulkner (1897–1962)
“When the history of this period is written, [William Jennings] Bryan will stand out as one of the most remarkable men of his generation and one of the biggest political men of our country.”
—William Howard Taft (1857–1930)