A theorem in finite group theory asserting that any finite group whose order is divisible by at most two distinct primes, i.e. of the form p^a q^b, is a solvable group.

Principle

Restricting the prime factorization of the group order to at most two distinct primes forces structural constraints (via Sylow subgroups, actions on cosets, and character-theoretic or counting arguments) that preclude the existence of nonabelian simple composition factors, yielding solvability.

Demonstration

Concrete instance: any group of order 18 = 2 · 3^2 must be solvable; an analysis of Sylow-3 and Sylow-2 subgroups and their normalizers shows that a nontrivial normal series with abelian factors exists.

Misapplication

Applying the theorem to groups whose order involves three distinct primes or to infinite groups; e.g. concluding solvability for a group of order pqr without verifying the specific combinatorial constraints is invalid.

Consequence

When applicable, the theorem rules out nonabelian simple groups of those orders and allows one to build explicit normal series, which simplifies classification and representation problems for such finite groups.

Reversal

The statement 'every solvable finite group has order p^a q^b' is false: solvability does not imply the order has at most two prime divisors, so inverting the theorem yields a misleading criterion.

Boundary

Applies only to finite groups and to orders with at most two distinct prime divisors; it does not cover orders with three or more distinct primes, nor does it directly address group structure beyond solvability (for instance, it does not classify all solvable groups of that order).

Semantic Tension

Often confused with more general Burnside results or with results that require additional hypotheses (for example versions using character theory); the tension is between the simple numeric hypothesis on order and deeper representation-theoretic proofs behind solvability.

Synthesis

Burnside's P^a Q^b theorem ties a purely arithmetical restriction on group order to a decisive structural conclusion: restricting distinct prime divisors to two forces enough normal subgroup structure to ensure the group is solvable.