Burnside's theorem is an early milestone of finite group theory: if the group order involves only two primes, with arbitrary exponents, the group is solvable. [@isaacs2008]
Definition
It is called Burnside's \(p^a q^b\) theorem and should not be confused with the Burnside basis theorem or normal-complement theorem.
Core result
Burnside's original proof used characters. The source develops a group-theoretic route from local normal complements and Thompson methods: proper local subgroups of a minimal counterexample are solvable, and normal-complement/minimal-normal analysis creates a nontrivial solvable normal layer, a contradiction.
Structural properties
- The order of a nonabelian simple group has at least three distinct prime divisors.
- The conclusion is solvability, not nilpotence or supersolvability.
- \(S_3\) of order \(2\cdot3\) is solvable but not nilpotent.
- \(A_4\) of order \(2^2\cdot3\) is solvable with nonnormal Sylow subgroups.
- Hall theory provides further prime-set structure inside solvable groups.
The theorem converts a purely arithmetic restriction into termination of the global derived series and is a model result on order controlling structure.
Example and boundary
Knowledge network
Solvable groups explain the conclusion, Hall theory decomposes prime sets internally, and Frobenius/Thompson tools enter the group-theoretic proof.
Proof and sources
Isaacs, Theorem 7.8 completes the proof built from the preceding local tools; the historical proof was character-theoretic. [@isaacs2008]
