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Lagrange, Joseph Louis, Comte, 1736-1813, French mathematician and astronomer, b. Turin, of French and Italian descent. Before the age of 20 he was professor of geometry at the royal artillery school at Turin. With his pupils he organized (1759) a society from which the Turin Academy of Sciences developed. Among his early successes were his method of solving isoperimetrical problems, on which the calculus of variations is based in part; his researches on the nature and propagation of sound and on the vibration of strings; and his studies on the libration of the moon and on the satellites of Jupiter. On the recommendation of Euler and D'Alembert, Frederick the Great invited him (1766) to succeed Euler as director of mathematics at the Berlin Academy of Sciences. The memoirs of the academy were enriched by his distinguished treatises, and during this time he wrote his chief work, *Mécanique analytique,* a treatment of mechanics based solely on algebra and the calculus and containing not a single diagram or geometric explanation. This was published (1788) in Paris, where he had been called by Louis XVI in 1787. In 1793 he became president of the commission on weights and measures; he was influential in causing the adoption of the decimal base for the metric system. A professor at the École polytechnique from 1797, he developed the use in teaching of the analytic method that he so skillfully employed in his research. He wrote *Théorie des fonctions analytiques* (1797) and *Leçons sur le calcul des fonctions* (1806), both based on his lectures. Under Napoleon, Lagrange was made senator and count; he is buried in the Panthéon. His contributions to the development of mathematics also include the application of differential calculus to the theory of probabilities and notable work on the solution of equations. In astronomy he is known for his calculations of the motions of planets.

The Columbia Electronic Encyclopedia Copyright © 2004.

Licensed from Columbia University Press

Licensed from Columbia University Press

Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order (number of elements) of every subgroup H of G divides the order of G. Lagrange's theorem is named after Joseph Lagrange.

This proof also shows that the quotient of the orders |G| / |H| is equal to the index [G : H] (the number of left cosets of H in G). If we write this statement as

- |G| = [G : H] · |H|,

then, interpreted as a statement about cardinal numbers, it remains true even for infinite groups G and H.

- a
^{n}= e.

This can be used to prove Fermat's little theorem and its generalization, Euler's theorem. These special cases were known long before the general theorem was proved.

The theorem also shows a group of prime order is cyclic and simple.

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Last updated on Thursday October 02, 2008 at 21:10:54 PDT (GMT -0700)

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