Feit–Thompson theorem

Every finite group of odd order is solvable. This theorem, proved by Walter Feit and John G. Thompson, established a central structural result in finite group theory.

Connect

Finite group: A group with finitely many elements. The theorem applies specifically to groups with finitely many elements.
Minimal counterexample: A counterexample chosen minimal under a specified measure, so smaller cases satisfy the claim. The proof analyzes a smallest hypothetical nonsolvable group of odd order.
Walter Feit: An Austrian-born American mathematician who worked in finite group theory. Feit co-authored the proof and shared responsibility for its extensive argument.
Odd order theorem: Another name for the Feit–Thompson theorem, asserting that every finite group of odd order is solvable. The theorem is also commonly known by this shorter name.
Cyclic group of order two: The group with two elements, one of which has order two. It is nonsolvable only in no meaningful sense, but demonstrates that even order is outside the theorem’s hypothesis.
Group order: The number of elements in a finite group. The hypothesis concerns whether this number is odd.
Sylow theorems: Theorems describing subgroups whose orders are maximal powers of a prime dividing a finite group’s order. Odd-order groups have Sylow subgroups for odd primes, whose structure constrains the proof.
John G. Thompson: An American mathematician whose work transformed the study of finite groups. Thompson co-authored the theorem and received the Fields Medal partly for related work.
Nonabelian simple group: A simple group that is not commutative. No finite nonabelian simple group can have odd order, by the theorem.
Alternating group A5: The nonabelian simple group of even order 60 consisting of even permutations of five symbols. It is a concrete nonsolvable finite group, and its even order avoids the theorem’s conclusion.