Questions: Burnside's Theorem

Question 1

Burnside's theorem states that groups of order pᵃqᵇ are solvable. Why can't this be proved by purely group-theoretic methods (without representation theory)?

• A. It has been proved by purely group-theoretic methods — the representation-theoretic proof was just the first one found
• B. No purely group-theoretic proof is known; the character-theoretic argument remains essential
• C. It can be proved using Sylow theorems alone
• D. The theorem is actually false without the representation-theoretic hypothesis

Answer: (A)

A purely group-theoretic proof was eventually found by Goldschmidt (1970) and simplified by others, but it is significantly more complex than Burnside's character-theoretic proof. For decades after Burnside's 1904 proof, no alternative was known. The representation-theoretic proof remains the standard one taught in algebra courses because of its elegance and the insight it provides into why character theory is powerful.

Question 2

The key lemma in Burnside's proof states: if χ is an irreducible character of degree d and g is an element whose conjugacy class has size coprime to d, then |χ(g)| = d or χ(g) = 0. This uses the fact that χ(g)/d is an algebraic integer.

• T. True
• F. False

Answer: True

The argument shows that χ(g)/d is both an algebraic integer (using the column orthogonality relations and the coprimality condition) and has absolute value at most 1 (since χ(g) is a sum of d roots of unity). An algebraic integer of absolute value ≤ 1 that is also a root-of-unity sum must be 0 or have absolute value exactly 1, giving |χ(g)| = 0 or d. This number-theoretic constraint is what forces the group to be solvable.

Question 3

A group of order 12 = 2² · 3 is solvable by Burnside's theorem. Name one group of order 12 and verify it is solvable.

Model answer: A₄ (the alternating group on 4 elements) has order 12. Its normal subgroup V₄ = {e, (12)(34), (13)(24), (14)(23)} is the Klein 4-group, with A₄/V₄ ≅ ℤ/3ℤ. Both V₄ and ℤ/3ℤ are abelian, so A₄ has an abelian normal series: {e} ◁ V₄ ◁ A₄, confirming solvability.

Every group of order pᵃqᵇ is solvable, so this includes groups of order 12 = 2²·3, order 24 = 2³·3, order 100 = 2²·5², etc. The smallest non-solvable group is A₅ of order 60 = 2²·3·5, which involves three prime factors — just beyond the reach of Burnside's theorem.