Burnside's Theorem

Burnside's theorem states that every group of order ^a q^b$ — where $ and $ are prime numbers — is solvable. Proved by William Burnside in 1904 using techniques from character theory, it was a precursor to the much more general Feit-Thompson theorem, which extends the solvability result to all groups of odd order.

Burnside's theorem was one of the first major applications of character theory to finite group theory, and it established the pattern that local constraints on group order can imply global structural properties. The theorem remains a standard result in graduate algebra courses and a stepping stone toward the classification of finite simple groups.

Burnside's theorem is often taught as a historical curiosity — a stepping stone to greater things. But the pattern it established, that the arithmetic of group order constrains its algebraic structure, is the deepest insight in all of finite group theory. The Feit-Thompson theorem is just Burnside's theorem with the training wheels removed.