Classification of Finite Simple Groups - History of The Proof - Timeline of The Proof

Timeline of The Proof

Many of the items in the list below are taken from Solomon (2001). The date given is usually the publication date of the complete proof of a result, which is sometimes several years later than the proof or first announcement of the result, so some of the items appear in the "wrong" order.

Publication date
1832 Galois introduces normal subgroups and finds the simple groups An (n ≥ 5) and PSL2(Fp) (p ≥ 5)
1854 Cayley defines abstract groups
1861 Mathieu describes the first two Mathieu groups M11, M12, the first sporadic simple groups, and announces the existence of M24.
1870 Jordan lists some simple groups: the alternating and projective special linear ones, and emphasizes the importance of the simple groups.
1872 Sylow proves the Sylow theorems
1873 Mathieu introduces three more Mathieu groups M22, M23, M24.
1892 Otto Hölder proves that the order of any nonabelian finite simple group must be a product of at least 4 primes, and asks for a classification of finite simple groups.
1893 Cole classifies simple groups of order up to 660
1896 Frobenius and Burnside begin the study of character theory of finite groups.
1899 Burnside classifies the simple groups such that the centralizer of every involution is a non-trivial elementary abelian 2-group.
1901 Frobenius proves that a Frobenius group has a Frobenius kernel, so in particular is not simple.
1901 Dickson defines classical groups over arbitrary finite fields, and exceptional groups of type G2 over fields of odd characteristic.
1901 Dickson introduces the exceptional finite simple groups of type E6.
1904 Burnside uses character theory to prove Burnside's theorem that the order of any non-abelian finite simple group must be divisible by at least 3 distinct primes.
1905 Dickson introduces simple groups of type G2 over fields of even characteristic
1911 Burnside conjectures that every non-abelian finite simple group has even order
1928 Hall proves the existence of Hall subgroups of solvable groups
1933 Hall begins his study of p-groups
1935 Brauer begins the study of modular characters.
1936 Zassenhaus classifies finite sharply 3-transitive permutation groups
1938 Fitting introduces the Fitting subgroup and proves Fitting's theorem that for solvable groups the Fitting subgroup contains its centralizer.
1942 Brauer describes the modular characters of a group divisible by a prime to the first power.
1954 Brauer classifies simple groups with GL2(Fq) as the centralizer of an involution.
1955 The Brauer–Fowler theorem implies that the number of finite simple groups with given centralizer of involution is finite, suggesting an attack on the classification using centralizers of involutions.
1955 Chevalley introduces the Chevalley groups, in particular introducing exceptional simple groups of types F4, E7, and E8.
1956 Hall–Higman theorem
1957 Suzuki shows that all finite simple CA groups of odd order are cyclic.
1958 The Brauer–Suzuki–Wall theorem characterizes the projective special linear groups of rank 1, and classifies the simple CA groups.
1959 Steinberg introduces the Steinberg groups, giving some new finite simple groups, of types 3D4 and 2E6 (the latter were independently found at about the same time by Jacques Tits).
1959 The Brauer–Suzuki theorem about groups with generalized quaternion Sylow 2-subgroups shows in particular that none of them are simple.
1960 Thompson proves that a group with a fixed-point-free automorphism of prime order is nilpotent.
1960 Feit, Hall, and Thompson show that all finite simple CN groups of odd order are cyclic.
1960 Suzuki introduces the Suzuki groups, with types 2B2.
1961 Ree introduces the Ree groups, with types 2F4 and 2G2.
1963 Feit and Thompson prove the odd order theorem.
1964 Tits introduces BN pairs for groups of Lie type and finds the Tits group
1965 The Gorenstein–Walter theorem classifies groups with a dihedral Sylow 2-subgroup.
1966 Glauberman proves the Z* theorem
1966 Janko introduces the Janko group J1, the first new sporadic group for about a century.
1968 Glauberman proves the ZJ theorem
1968 Higman and Sims introduce the Higman–Sims group
1968 Conway introduces the Conway groups
1969 Walter's theorem classifies groups with abelian Sylow 2-subgroups
1969 Introduction of the Suzuki sporadic group, the Janko group J2, the Janko group J3, the McLaughlin group, and the Held group.
1969 Gorenstein introduces signalizer functors based on Thompson's ideas.
1970 Bender introduced the generalized Fitting subgroup
1970 The Alperin–Brauer–Gorenstein theorem classifies groups with quasi-dihedral or wreathed Sylow 2-subgroups, completing the classification of the simple groups of 2-rank at most 2
1971 Fischer introduces the three Fischer groups
1971 Thompson classifies quadratic pairs
1971 Bender classifies group with a strongly embedded subgroup
1972 Gorenstein proposes a 16-step program for classifying finite simple groups; the final classification follows his outline quite closely.
1972 Lyons introduces the Lyons group
1973 Rudvalis introduces the Rudvalis group
1973 Fischer discovers the baby monster group (unpublished), which Fischer and Griess use to discover the monster group, which in turn leads Thompson to the Thompson sporadic group and Norton to the Harada–Norton group (also found in a different way by Harada).
1974 Thompson classifies N-groups, groups all of whose local subgroups are solvable.
1974 The Gorenstein–Harada theorem classifies the simple groups of sectional 2-rank at most 4, dividing the remaining finite simple groups into those of component type and those of characteristic 2 type.
1974 Tits shows that groups with BN pairs of rank at least 3 are groups of Lie type
1974 Aschbacher classifies the groups with a proper 2-generated core
1975 Gorenstein and Walter prove the L-balance theorem
1976 Glauberman proves the solvable signalizer functor theorem
1976 Aschbacher proves the component theorem, showing roughly that groups of odd type satisfying some conditions have a component in standard form. The groups with a component of standard form were classified in a large collection of papers by many authors.
1976 O'Nan introduces the O'Nan group
1976 Janko introduces the Janko group J4, the last sporadic group to be discovered
1977 Aschbacher characterizes the groups of Lie type of odd characteristic in his classical involution theorem. After this theorem, which in some sense deals with "most" of the simple groups, it was generally felt that the end of the classification was in sight.
1978 Timmesfeld proves the O2 extraspecial theorem, breaking the classification of groups of GF(2)-type into several smaller problems.
1978 Aschbacher classifies the thin finite groups, which are mostly rank 1 groups of Lie type over fields of even characteristic.
1981 Bombieri uses elimination theory to complete Thompson's work on the characterization of Ree groups, one of the hardest steps of the classification.
1982 McBride proves the signalizer functor theorem for all finite groups.
1982 Griess constructs the monster group by hand
1983 The Gilman–Griess theorem classifies groups groups of characteristic 2 type and rank at least 4 with standard components, one of the three cases of the trichotomy theorem.
1983 Aschbacher proves that no finite group satisfies the hypothesis of the uniqueness case, one of the three cases given by the trichotomy theorem for groups of characteristic 2 type.
1983 Gorenstein and Lyons prove the trichotomy theorem for groups of characteristic 2 type and rank at least 4, while Aschbacher does the case of rank 3. This divides these groups into 3 subcases: the uniqueness case, groups of GF(2) type, and groups with a standard component.
1983 Gorenstein announces the proof of the classification is complete, somewhat prematurely as the proof of the quasithin case was incomplete.
1994 Gorenstein, Lyons, and Solomon begin publication of the revised classification
2004 Aschbacher and Smith publish their work on quasithin groups (which are mostly groups of Lie type of rank at most 2 over fields of even characteristic), filling the last gap in the classification known at that time.
2008 Harada and Solomon fill a minor gap in the classification by describing groups with a standard component that is a cover of the Mathieu group M22, a case that was accidentally omitted from the proof of the classification due to an error in the calculation of the Schur multiplier of M22.

Read more about this topic:  Classification Of Finite Simple Groups, History of The Proof

Famous quotes containing the word proof:

    Sculpture and painting are very justly called liberal arts; a lively and strong imagination, together with a just observation, being absolutely necessary to excel in either; which, in my opinion, is by no means the case of music, though called a liberal art, and now in Italy placed even above the other two—a proof of the decline of that country.
    Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)