Lagrange's theorem (group theory)

Lagrange's theorem (group theory)

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.

Proof of Lagrange's Theorem

This can be shown using the concept of left cosets of "H" in "G". The left cosets are the equivalence classes of a certain equivalence relation on "G" and therefore form a partition of "G". If we can show that all cosets of "H" have the same number of elements, then we are done, since "H" itself is a coset of "H". Now, if "aH" and "bH" are two left cosets of "H", we can define a map "f" : "aH" → "bH" by setting "f"("x") = "ba"-1"x". This map is bijective because its inverse is given by "f" −1("y") = "ab"−1"y".

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".

Using the theorem

A consequence of the theorem is that the order of any element "a" of a finite group (i.e. the smallest positive integer "k" with "a""k" = "e") divides the order of that group, since the order of "a" is equal to the order of the cyclic subgroup generated by "a". If the group has "n" elements, it follows

:"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.

Existence of subgroups of a certain order

Lagrange's theorem raises the question of whether every divisor of the order of a group is the order of a subgroup. This need not hold. Given a finite group "G" and a divisor "d" of |"G"|, there does not necessarily exist a subgroup of "G" with order "d". The smallest example is the alternating group "G" = "A"4 which has 12 elements but no subgroup of order 6. Any finite group which has a subgroup with size equal to any (positive) divisor of the size of the group must be solvable, so nonsolvable groups are examples of this phenonenon, although "A"4 shows that they aren't the only examples. If "G" is abelian, then there always exists a subgroup of any order dividing the size of "G". A partial generalization is given by Cauchy's theorem.

History

Lagrange did not prove Lagrange's theorem in its general form. What he actually proved was 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 3 is a factor of 6.)The number of such polynomials is the index in the symmetric group "S"n of the subgroup "H" of permutations which preserve the polynomial. (For the example of "x" + "y" − "z", the subgroup "H" in "S"3 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.

See also

* Sylow's theorem

References

*
* | year=2004


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Cauchy's theorem (group theory) — Cauchy s theorem is a theorem in the mathematics of group theory, named after Augustin Louis Cauchy. It states that if G is a finite group and p is a prime number dividing the order of G (the number of elements in G ), then G contains an element… …   Wikipedia

  • Lagrange's theorem — In mathematics, Lagrange s theorem usually refers to any of the following theorems, attributed to Joseph Louis Lagrange:* Lagrange s theorem (group theory) * Lagrange s theorem (number theory) * Lagrange s four square theorem, which states that… …   Wikipedia

  • Group theory — is a mathematical discipline, the part of abstract algebra that studies the algebraic structures known as groups. The development of group theory sprang from three main sources: number theory, theory of algebraic equations, and geometry. The… …   Wikipedia

  • Lagrange inversion theorem — In mathematical analysis, the Lagrange inversion theorem, also known as the Lagrange Bürmann formula, gives the Taylor series expansion of the inverse function of an analytic function. Theorem statementSuppose the dependence between the variables …   Wikipedia

  • History of group theory — The history of group theory, a mathematical domain studying groups in their various forms, has evolved in various parallel threads. There are three historical roots of group theory: the theory of algebraic equations, number theory and geometry.… …   Wikipedia

  • Elementary group theory — In mathematics, a group is defined as a set G and a binary operation on G , called product and denoted by infix * . Product obeys the following rules (also called axioms). Let a , b , and c be arbitrary elements of G . Then: *A1, Closure. a * b… …   Wikipedia

  • List of group theory topics — Contents 1 Structures and operations 2 Basic properties of groups 2.1 Group homomorphisms 3 Basic types of groups …   Wikipedia

  • Glossary of group theory — A group ( G , •) is a set G closed under a binary operation • satisfying the following 3 axioms:* Associativity : For all a , b and c in G , ( a • b ) • c = a • ( b • c ). * Identity element : There exists an e ∈ G such that for all a in G , e •… …   Wikipedia

  • Order (group theory) — This article is about order in group theory. For order in other branches of mathematics, see Order (mathematics). For order in other disciplines, see Order. In group theory, a branch of mathematics, the term order is used in two closely related… …   Wikipedia

  • Lagrange (disambiguation) — Lagrange may refer to: * Château Lagrange, the wine from Bordeaux, France * Joseph Louis Lagrange, (1736–1813) mathematician and mathematical physicist * Léo Lagrange, (1900–1940) french ministre * Georges Lagrange, (1928–2004) esperanto writerIn …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”