Burnside's Theorem

Core Idea

Burnside's pᵃqᵇ theorem states that every group of order pᵃqᵇ (for primes p, q) is solvable. The proof, remarkably, uses representation theory — specifically, the fact that a character of degree d evaluated at an element whose conjugacy class has size coprime to d must be zero or have absolute value d. This was one of the first major applications of character theory to pure group theory, demonstrating that representation-theoretic methods can prove statements with no obvious connection to linear algebra.

Explainer

Burnside's theorem (1904) is one of the crown jewels of character theory. It states: every group of order pᵃqᵇ, where p and q are primes, is solvable — meaning it can be built up from abelian groups through a series of normal subgroup extensions. The remarkable feature is not the result itself (which had been conjectured on empirical grounds) but the method of proof: it uses character theory in an essential way, deploying representation-theoretic tools to prove a statement about abstract group structure.

The proof hinges on a lemma about characters and conjugacy class sizes. Suppose χ is an irreducible character of degree d and g ∈ G has a conjugacy class of size m, where gcd(d, m) = 1. Then χ(g)/d is an algebraic integer (this follows from the column orthogonality relations and the coprimality condition via a Bezout identity argument). But χ(g) is a sum of d roots of unity, so |χ(g)| ≤ d, meaning |χ(g)/d| ≤ 1. An algebraic integer with all conjugates of absolute value ≤ 1 is either zero or a root of unity with absolute value 1. Therefore |χ(g)| = 0 or |χ(g)| = d.

The conclusion |χ(g)| = d forces χ(g) = d · (root of unity), which means ρ(g) is a scalar matrix — so g lies in the center of ρ(G). Using this, Burnside shows that a group of order pᵃqᵇ always has a nontrivial proper normal subgroup (by finding a conjugacy class whose size is a prime power and applying the lemma). Induction on |G| then gives solvability.

This theorem illustrates a paradigm that recurs throughout algebra: representation-theoretic methods can prove results about groups that seem inaccessible by direct group-theoretic arguments. The interplay between the number-theoretic properties of characters (algebraic integers, roots of unity) and the combinatorial structure of the group (conjugacy class sizes, subgroup lattice) creates leverage that purely combinatorial arguments lack. The Feit-Thompson theorem (1963) — all groups of odd order are solvable — extends this paradigm to its most spectacular conclusion, using character theory and modular representation theory in a 255-page proof.