Burnside’s Theorem

Theorem 1 (Burnside’s Theorem).

Let G be a simple groupMathworldPlanetmathPlanetmath, σG. Then the number of conjugatesPlanetmathPlanetmath of σ is not a prime power (unless σ is its own conjugacy classMathworldPlanetmath).

Proofs of this theorem are quite difficult and rely on representation theory.

From this we immediately get

Corollary 1.

A group G of order paqb, where p,q are prime, cannot be a nonabelianPlanetmathPlanetmath simple group.


Suppose it is. Then the center of G is trivial, {e}, since the center is a normal subgroupMathworldPlanetmath and G is simple nonabelian. So if Ci are the nontrivial conjugacy classes, we have from the class equationMathworldPlanetmathPlanetmath that


Now, each |Ci| divides |G|, but cannot be 1 since the center is trivial. It cannot be a power of either p or q by Burnside’s theorem. Thus pq|Ci| for each i and thus |G|1(modpq), which is a contradictionMathworldPlanetmathPlanetmath. ∎

Finally, a corollary of the above is known as the Burnside p-q Theorem (http://planetmath.org/BurnsidePQTheorem).

Corollary 2.

A group of order paqb is solvable.

Title Burnside’s Theorem
Canonical name BurnsidesTheorem
Date of creation 2013-03-22 16:38:14
Last modified on 2013-03-22 16:38:14
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 4
Author rm50 (10146)
Entry type Theorem
Classification msc 20D05