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[ | {{For|the counting result sometimes called "Burnside's theorem"|Burnside's lemma}} | ||
In [[mathematics]], '''Burnside's theorem''' in [[group theory]] states that if ''G'' is a [[finite group]] of [[Order (group theory)|order]] | |||
:<math>p^a q^b\ </math> | |||
where ''p'' and ''q'' are [[prime number]]s, and ''a'' and ''b'' are [[negative and positive numbers|non-negative]] [[integer]]s, then ''G'' is [[Solvable group|solvable]]. Hence each | |||
non-Abelian [[finite simple group]] has order divisible by at least three distinct primes. | |||
==History== | |||
The theorem was proved by [[William Burnside]] in the early years of the 20th century. | |||
Burnside's theorem has long been one of the best-known applications of [[Representation theory of finite groups|representation theory]] to the theory of finite groups, though a proof avoiding the use of group characters was published by D. Goldschmidt around 1970. | |||
==Outline of Burnside's proof== | |||
# By induction, it suffices to prove that a finite simple group ''G'' whose order has the form <math>p^a q^b </math> for primes ''p'' and ''q'' is cyclic. Suppose then that the order of ''G'' has this form, but ''G'' is not cyclic. Suppose for definiteness that ''b'' > 0. | |||
# Using the modified class equation, ''G'' has a non-identity conjugacy class of size prime to ''q''. Hence ''G'' either has a non-trivial [[Group center|center]], or has a [[conjugacy class]] of size <math>p^r</math> for some positive integer ''r''. The first possibility is excluded since ''G'' is assumed simple, but not cyclic. Hence there is a non-central element ''x'' of ''G'' such that the conjugacy class of ''x'' has size <math>p^r</math>. | |||
# Application of [[Character theory|column orthogonality relations]] and other properties of group characters and algebraic integers lead to the existence of a non-trivial [[Character theory|irreducible character]] <math>\chi</math> of ''G'' such that <math>|\chi(x)| = \chi(1)</math>. | |||
# The simplicity of ''G'' then implies that any non-trivial complex irreducible representation is faithful, and it follows that ''x'' is in the center of ''G'', a contradiction. | |||
==References== | |||
# James, Gordon; and Liebeck, Martin (2001). ''Representations and Characters of Groups'' (2nd ed.). Cambridge University Press. ISBN 0-521-00392-X. See chapter 31. | |||
# Fraleigh, John B. (2002) ''A First Course in Abstract Algebra'' (7th ed.). Addison Wesley. ISBN 0-201-33596-4. | |||
[[Category:Theorems in group theory]] | |||
Revision as of 22:16, 7 January 2014
28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance. In mathematics, Burnside's theorem in group theory states that if G is a finite group of order
where p and q are prime numbers, and a and b are non-negative integers, then G is solvable. Hence each non-Abelian finite simple group has order divisible by at least three distinct primes.
History
The theorem was proved by William Burnside in the early years of the 20th century.
Burnside's theorem has long been one of the best-known applications of representation theory to the theory of finite groups, though a proof avoiding the use of group characters was published by D. Goldschmidt around 1970.
Outline of Burnside's proof
- By induction, it suffices to prove that a finite simple group G whose order has the form for primes p and q is cyclic. Suppose then that the order of G has this form, but G is not cyclic. Suppose for definiteness that b > 0.
- Using the modified class equation, G has a non-identity conjugacy class of size prime to q. Hence G either has a non-trivial center, or has a conjugacy class of size for some positive integer r. The first possibility is excluded since G is assumed simple, but not cyclic. Hence there is a non-central element x of G such that the conjugacy class of x has size .
- Application of column orthogonality relations and other properties of group characters and algebraic integers lead to the existence of a non-trivial irreducible character of G such that .
- The simplicity of G then implies that any non-trivial complex irreducible representation is faithful, and it follows that x is in the center of G, a contradiction.
References
- James, Gordon; and Liebeck, Martin (2001). Representations and Characters of Groups (2nd ed.). Cambridge University Press. ISBN 0-521-00392-X. See chapter 31.
- Fraleigh, John B. (2002) A First Course in Abstract Algebra (7th ed.). Addison Wesley. ISBN 0-201-33596-4.