Burnside's theorem: Difference between revisions

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In mathematics, Burnside's theorem in group theory states that if G is a finite group of order

p^{a}q^{b}

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 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 application 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

Using mathematical induction, it suffices to prove that a simple group G whose order has this form is Abelian, so the proof begins by assuming that G is simple group of order $p^{a}q^{b}$ , and aims to prove that G is Abelian.
Using Sylow's theorem , G either has a non-trivial center, or has a conjugacy class of size $p^{r}$ for some integer r ≥ 1. In the former case, G must be Abelian, by its simplicity, so it may be assumed that there is an element x of G such that the conjugacy class of x has size $p^{r}$ > 1.
Application of column orthogonality relations and properties of algebraic integers lead to the to the existence of a non-trivial irreducible character $\chi$ of G such that $|\chi (x)|=\chi (1)$ .
The simplicity of G implies that any complex irreducible representation with character $\chi$ is faithful, and it follows that x is in the center of G, contrary to the fact that the size of the conjugacy class has size greater than 1.

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.

@@ Line 3: / Line 3: @@
 :<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 ''G'' is cyclic of prime order or is not [[Simple group|simple]]).
+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 three distinct primes.
 ==History==
-The theorem was stated by [[William Burnside]].
+The theorem was proved by [[William Burnside]] in the early years of the 20th century.
-Burnside's theorem is a well-known application of [[Group representation|representation theory]] to the theory of finite groups because it wasn't until after 1970 that a proof that was not based on representation theory was given.
+Burnside's theorem has long been one of the best-known application of [[Group representation|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 the proof==
+==Outline of Burnside's proof==
-# It turns out that a group of order <math>p^a q^b</math> is either easily decomposable with [[Sylow theorems|Sylow theory]], has an easily recognizable non-trivial [[Group center|center]], or has a [[conjugacy class]] of order <math>p^r</math> for some integer ''r'' ≥ 1.
-# If we have a conjugacy class of order <math>p^r</math>, then there is a [[Group representation|representation]] ρ of ''G'' that either has a [[Subgroup|proper]] [[Subgroup|non-trivial]] [[Group representation#Basic definitions|kernel]] or is [[Group representation#Basic definitions|faithful]], in which case it will follow that the [[center of a group|center]] of ''G'' is non-trivial.
-# To build the representation ρ given a conjugacy class <math>g^G</math> with representative ''g'' and with order <math>p^r</math>, we apply the [[Character theory|column orthogonality relations]] to the [[Character theory|character table]] of ''G'' to get an equality which we fiddle with algebraically to demonstrate the existence of an [[Character theory|irreducible character]] <math>\chi_i</math> of ''G'' such that <math>\chi_i(g) / \chi_i(1)</math> is not an [[algebraic integer]]. We subsequently find that <math>\chi_i(g) / \chi_i(1)</math> is [[coprime]] to <math>p^r</math>.
-# We attack from a different angle by showing, with an extensive bit of further algebraic manipulation and representation theory, that the [[Conjugacy class sum|class sum]] <math>\overline{C}</math> of the conjugacy class <math>g^G</math> in the [[group algebra]] <math>\mathbb{C} G</math> is equal (in its action on the group algebra) to an algebraic integer λ.
-# Substituting λ back into previous work and applying a little bit more representation theory demonstrates that if ρ is a representation of ''G'' with character <math>\chi_i</math> then either the kernel of ρ is a proper non-trivial [[normal subgroup]] of ''G'' or ρ is faithful in which case ''g'', the representative of the conjugacy class that we started with, is in the center of ''G'', which is therefore non-trivial. In either case ''G'' is not simple.
+# Using [[mathematical induction]], it suffices to prove that a simple group ''G'' whose order has this form is Abelian, so the proof begins by assuming that ''G'' is simple group of order <math>p^a q^b</math>, and aims to prove that ''G'' is Abelian.
+# Using [[Sylow theorems|Sylow's theorem ]], ''G'' either has a non-trivial [[Group center|center]], or has a [[conjugacy class]] of size <math>p^r</math> for some integer ''r'' ≥ 1. In the former case, ''G'' must be Abelian, by its simplicity, so it may be assumed that there is an element ''x'' of ''G'' such that the conjugacy class of ''x'' has size <math>p^r</math> > 1.
+# Application of [[Character theory|column orthogonality relations]] and properties of [[algebraic integers]] lead to the  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'' implies that any complex irreducible representation with character <math>\chi</math> is faithful, and it follows that ''x'' is in the center of ''G'', contrary to the fact that the size of the conjugacy class has size greater than ''1''.
 ==References==