Burnside's Theorem

Burnside's theorem uses character theory to prove remarkable results in pure group theory, demonstrating the power of representation-theoretic methods beyond their original domain.

TheoremBurnside's $p^a q^b$ Theorem

Let $G$ be a finite group whose order is $|G| = p^a q^b$ where $p, q$ are primes and $a, b \geq 0$ . Then $G$ is solvable.

This theorem, proved using character theory, was a major achievement with no known purely group-theoretic proof at the time of its discovery (1904).

The proof uses the following key lemma about algebraic integers and class equations.

TheoremBurnside's Class Equation Lemma

Let $\chi$ be the character of a non-trivial irreducible representation of $G$ over $\mathbb{C}$ , and let $g \in G$ be an element whose conjugacy class has size $|C_g|$ coprime to $\dim \chi$ . Then $\chi(g) = 0$ .

Proof idea: The value $\frac{|C_g| \chi(g)}{\dim \chi}$ is an algebraic integer (by properties of character values). If $\gcd(|C_g|, \dim \chi) = 1$ , then $\frac{\chi(g)}{\dim \chi}$ is an algebraic integer. But $|\chi(g)| \leq \dim \chi$ with equality only if $g$ is in the center. Combining these constraints forces $\chi(g) = 0$ .

ExampleApplication to Small Groups

Groups of order $p^a$ (prime power): These are always solvable (in fact, nilpotent). This follows from basic group theory without characters.

Groups of order $pq$ (two distinct primes): By Burnside's theorem, all such groups are solvable. Explicitly:

If $p < q$ and $q \not\equiv 1 \pmod{p}$ , then $G \cong \mathbb{Z}_{pq}$ (cyclic)
Otherwise, $G$ is a semidirect product $\mathbb{Z}_q \rtimes \mathbb{Z}_p$

No simple groups of order $p^a q^b$ except cyclic of prime order: Burnside's theorem immediately implies that non-abelian finite simple groups must have at least three distinct prime divisors.

TheoremBurnside's Counting Lemma (Burnside's Lemma)

Let $G$ act on a finite set $X$ . The number of orbits is: $\text{# orbits} = \frac{1}{|G|} \sum_{g \in G} |X^g|$ where $X^g = \{x \in X : g \cdot x = x\}$ is the fixed point set of $g$ .

This is equivalent to: the average number of fixed points equals the number of orbits.

ExampleUsing Burnside's Lemma

Coloring a square: How many ways to color the vertices of a square with 2 colors, up to rotation?

The group $G = \mathbb{Z}/4\mathbb{Z}$ acts on $X = \{$ colorings $\}$ with $|X| = 2^4 = 16$ . The fixed points:

Identity: all 16 colorings fixed
Rotation by $90°$ : 2 colorings fixed (all same color)
Rotation by $180°$ : 4 colorings fixed (opposite pairs same)
Rotation by $270°$ : 2 colorings fixed (all same color)

Number of orbits: $\frac{1}{4}(16 + 2 + 4 + 2) = 6$ .

Remark

Burnside's $p^a q^b$ theorem remained the only general result on solvability for decades. The Feit-Thompson theorem (1963) proved that all groups of odd order are solvable, a monumental achievement requiring 255 pages and extensive character theory. Together, these results show that representation theory is not merely auxiliary to group theory but essential for proving deep structural theorems.