Burnside's `pᵃqᵇ` theorem

Every finite group whose order has at most two distinct prime factors is solvable. This is the sharpest classical result relating order arithmetic to structure — any non-abelian simple group has at least three distinct primes dividing its order.

Statement

Burnside (1904). Let G be a finite group with |G| = pᵃ qᵇ for primes p, q and integers a, b ≥ 0. Then G is solvable.

Consequences

It narrows the hunt for simple groups. If |G| has only two prime divisors, G has a non-trivial normal subgroup (unless |G| = p prime). Non-abelian simple groups must have ≥ 3 distinct prime divisors in their order. The smallest example is A₅ with |A₅| = 60 = 2² · 3 · 5.
It seeded Feit-Thompson. The conjecture "every group of odd order is solvable" was stated by Burnside in his 1911 book; proved by Feit and Thompson in 1963 — a 250+ page argument, a foundational step in the classification of finite simple groups.

Proof idea

The proof Burnside gave is character-theoretic. The key tool is

Burnside's lemma on zero columns. If χ is an irreducible character of G and K is a conjugacy class of size coprime to χ(1), then χ(g) = 0 or |χ(g)| = χ(1) for any g ∈ K.

Applied carefully (and combined with orthogonality relations and the integrality of character values), this lemma forces the existence of a non-trivial normal subgroup in any pᵃqᵇ-group. Induction then gives solvability.

A purely group-theoretic proof was found by Bender, Matsuyama, and Goldschmidt in the 1970s, but the character-theoretic route remains the standard textbook presentation.

Examples

Order 12 = 2² · 3. Every group of order 12 is solvable. The five groups are C₁₂, C₆ × C₂, D₆, A₄, and the dicyclic group Dic₃ (also known as Q₁₂).
Order 24 = 2³ · 3. All 15 groups of order 24 are solvable.
Order p² q² (e.g. 36, 100, 196) — always solvable.

Sharpness

The bound "two primes" is tight:

A₅, order 60 = 2² · 3 · 5 — simple, non-solvable. Three primes are the minimum.
|PSL(2,𝔽₇)| = 168 = 2³ · 3 · 7 — simple.
|PSL(2,𝔽₁₁)| = 660 = 2² · 3 · 5 · 11 — simple.

No finite simple non-abelian group has fewer than 3 distinct prime divisors in its order.