Classification theorem: every finite simple group is isomorphic to exactly one of: (1) a cyclic group of prime order; (2) an alternating group \(A_{n}\) (n ≥ 5); (3) a finite group of Lie type (including twisted forms); or (4) one of 26 sporadic groups.
Emerging from Galois's observation that A₅ is simple (1832), through Hölder's programme (1892), Burnside's pq-theorem (1904), Brauer's character theory (1945-76), Feit–Thompson's Odd Order Theorem (1963, 255 pages), Fischer's transposition groups, and the Monster's discovery by Fischer–Griess (1973/80). Gorenstein declared victory in 1983; the quasithin gap was filled by Aschbacher–Smith in 2004. The full proof exceeds 10,000 pages across hundreds of articles.
Completed 2004 (Aschbacher–Smith quasithin volume closing the final gap). The classification is unique in mathematics: no single person has read every line, yet the community accepts it based on distributed expertise and the second-generation simplification programme. Applications pervade every area of finite mathematics — from the Green–Tao theorem's model-theoretic prerequisites to the Moonshine conjectures connecting the Monster to modular forms. Second-generation proofs continue consolidating the edifice.