Burnside’s Theorem

Theorem 1 (Burnside’s Theorem).

Let $G$ be a simple group, $\sigma\in G$ . Then the number of conjugates of $\sigma$ is not a prime power (unless $\sigma$ is its own conjugacy class).

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

From this we immediately get

Corollary 1.

A group $G$ of order $p^{a}q^{b}$ , where $p, q$ are prime, cannot be a nonabelian simple group.

Proof.

Suppose it is. Then the center of $G$ is trivial, $\{e\}$ , since the center is a normal subgroup and $G$ is simple nonabelian. So if $C_{i}$ are the nontrivial conjugacy classes, we have from the class equation that

\lvert G\rvert=1+\sum\lvert C_{i}\rvert

Now, each $\lvert C_{i}\rvert$ divides $\lvert G\rvert$ , but cannot be $1$ since the center is trivial. It cannot be a power of either $p$ or $q$ by Burnside’s theorem. Thus $pq\mid\lvert C_{i}\rvert$ for each $i$ and thus $\lvert G\rvert\equiv 1\pmod{pq}$ , which is a contradiction. ∎

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

Corollary 2.

A group of order $p^{a}q^{b}$ 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