**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 A_{n} (n ≥ 5) and PSL_{2}(F_{p}) (p ≥ 5) |

1854 | Cayley defines abstract groups |

1861 | Mathieu describes the first two Mathieu groups M_{11}, M_{12}, the first sporadic simple groups, and announces the existence of M_{24}. |

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 M_{22}, M_{23}, M_{24}. |

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 G_{2} over fields of odd characteristic. |

1901 | Dickson introduces the exceptional finite simple groups of type E_{6}. |

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 G_{2} 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 GL_{2}(F_{q}) 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 F_{4}, E_{7}, and E_{8}. |

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 3D_{4} and 2E_{6} (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 2B_{2}. |

1961 | Ree introduces the Ree groups, with types 2F_{4} and 2G_{2}. |

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 O_{2} 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

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