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:
“No one is ahead of his time, it is only that the particular variety of creating his time is the one that his contemporaries who are also creating their own time refuse to accept.... For a very long time everybody refuses and then almost without a pause almost everybody accepts. In the history of the refused in the arts and literature the rapidity of the change is always startling.”
—Gertrude Stein (1874–1946)
“A country grows in history not only because of the heroism of its troops on the field of battle, it grows also when it turns to justice and to right for the conservation of its interests.”
—Aristide Briand (1862–1932)
“Literary works cannot be taken over like factories, or literary forms of expression like industrial methods. Realist writing, of which history offers many widely varying examples, is likewise conditioned by the question of how, when and for what class it is made use of.”
—Bertolt Brecht (1898–1956)