Burnside's p^a q^b Theorem

Burnside’s p^a q^b Theorem. Let $G$ be a finite group with

for primes $p,q$ and integers $a,b \ge 0$. Then $G$ is a solvable group .

This theorem is a landmark result: finiteness together with “at most two prime divisors” forces strong structural constraints. Standard proofs use tools from representation theory , especially properties of characters of finite groups, to produce a nontrivial normal subgroup and then argue by induction on $|G|$.

Proof sketch. One shows (using character theory) that $G$ has a nontrivial proper normal subgroup $N$. Then both $N$ and $G/N$ again have order of the form $p^{a'}q^{b'}$, so by induction they are solvable, and solvability is inherited by extensions.