Burnside’s Theorem

Theorem 1 (Burnside’s Theorem).

Let $G$ be a simple group, $\sigma \mathrm{\in }G$. 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 $p^a\mathit{}{q}^b$, where $p\mathrm{,}q$ 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

$$|G|=1+\sum |C_i|$$

Now, each $|C_i|$ 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 $pq\mid |C_i|$ for each $i$ and thus $|G|\equiv 1(modpq)$, 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 $p^a\mathit{}{q}^b$ is solvable.