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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
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==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''&nbsp;>&nbsp;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 23:59, 8 February 2014


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