Lagrange's Group Theorem

The most general form of Lagrange's group theorem, also known as Lagrange's lemma, states that for a group G, a subgroup H of G, and a subgroup K of H, (G:K)=(G:H)(H:K), where the products are taken as cardinalities (thus the theorem holds even for infinite groups) and (G:H) denotes the subgroup index for the subgroup H of G. A frequently stated corollary (which follows from taking K={e}, where e is the identity element) is that the order of G is equal to the product of the order of H and the subgroup index of H.

The corollary is easily proven in the case of G being a finite group, in which case the left cosets of H form a partition of G, thus giving the order of G as the number of blocks in the partition (which is (G:H)) multiplied by the number of elements in each partition (which is just the order of H).

For a finite group G, this corollary gives that the order of H must divide the order of G. Then, because the order of an element x of G is the order of the cyclic subgroup generated by x, we must have that the order of any element of G divides the order of G.

The converse of Lagrange's theorem is not, in general, true (Gallian 1993, 1994).

See also

Quotient Group

This entry contributed by Nicolas Bray

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Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra, 5th ed. New York: Macmillan, p. 111, 1996.Gallian, J. A. "On the Converse of Lagrange's Theorem." Math. Mag. 66, 23, 1993.Gallian, J. A. Contemporary Abstract Algebra, 3rd ed. Lexington, MA: D. C. Heath, 1994.Herstein, I. N. Abstract Algebra, 3rd ed. New York: Macmillan, p. 66, 1996.Hogan, G. T. "More on the Converse of Lagrange's Theorem." Math. Mag. 69, 375-376, 1996.Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 86, 1993.

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Lagrange's Group Theorem

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Bray, Nicolas. "Lagrange's Group Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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