Any finite group whose order is divisible by at most two distinct prime numbers is solvable. Proved by William Burnside using character theory, it was, before the Feit-Thompson theorem, the deepest known result on the solvability of finite groups.
Facts
StatementIf G is a finite group of order p^a q^b, where p and q are prime numbers and a and b are non-negative integers, then G is solvable. 1 Classification
Statement FormCharacterization Theorem 1 Connections
Sources
1. Burnside's Theorem (Wikipedia)
Wikimedia FoundationLead section, opening sentence
Burnside's theorem in group theory states that if G is a finite group of order p^a q^b where p and q are prime numbers, and a and b are non-negative integers, then G is solvable.
Lead section, attribution sentence
The theorem was proved by William Burnside (1904) using the representation theory of finite groups.
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