## A brief history of the classification of finite simple groups

Venue: | BAMS |

Citations: | 16 - 0 self |

### BibTeX

@ARTICLE{Solomon_abrief,

author = {Ronald Solomon},

title = {A brief history of the classification of finite simple groups},

journal = {BAMS},

year = {},

volume = {38},

pages = {315--352}

}

### OpenURL

### Abstract

Abstract. We present some highlights of the 110-year project to classify the finite simple groups.

### Citations

300 |
The theory of groups
- Hall
- 1959
(Show Context)
Citation Context ...rimes 2 and 3 are intertwined in the subgroup structure of G, Thompson’s more robust factorization approach returns to center stage. By 1959 when Marshall Hall published his text The Theory of Groups =-=[Ha]-=-, he could write in dramatic contrast to Brauer’s remarks in 1954: “Current research in Group Theory, as witnessed by the publications covered in Mathematical Reviews, is vigorous and extensive.” Hall... |

136 |
Finite Simple Groups
- Gorenstein
- 1982
(Show Context)
Citation Context ...sus on the desirability of replacing characteristic 2-type with some version of even-type as described here. There is much less agreement on the ideal definition of even-type. One version is given in =-=[GLS1]-=-. A different version (called “even characteristic” to avoid confusion) is given by Aschbacher and Smith. Yet another version has been proposed by Meierfrankenfeld. Groups of characteristic 2-type hav... |

127 |
Theory of Groups of Finite Order
- Burnside
- 1911
(Show Context)
Citation Context ...ast seven prime numbers. In a similar but more fruitful direction, Frobenius [Fr1] proved in 1893 that a simple group of squarefree order must be cyclic of prime order. Burnside extended this in 1895 =-=[Bu1]-=- to the following suggestive result: Theorem. If p is the smallest prime divisor of |G| and if G has a cyclic Sylow p-subgroup P ,thenG = KP, whereK is a normal subgroup of order prime to p. In partic... |

123 |
Linear groups with an exposition to the Galois field theory
- Dickson
- 1901
(Show Context)
Citation Context ... is completely reducible. Moore’s student, L.E. Dickson, extended Jordan’s database of simple groups to include all the classical projective groups over finite fields in his book Linear Groups (1900) =-=[D1]-=-. Dickson was well aware of the analogy between his work and the recent monumental results of Killing and Cartan classifying the simple continuous groups of Lie, and shortly thereafter (1901,1903) [D2... |

74 |
Solvability of groups of odd order
- Feit, Thomson
- 1963
(Show Context)
Citation Context ...urned out to be true, but very deep. Indeed the only known proof of the Schreier Conjecture is as a corollary of the Classification Theorem. The proof of the Odd Order Conjecture by Feit and Thompson =-=[FT]-=- finally yielded the unrestricted Schur-Zassenhaus Theorem, for which there is still no known elementary proof. In the same architectural spirit was the paper “Beiträge zur Theorie der endlichen Grupp... |

49 |
Sur certains groupes simples
- Chevalley
- 1955
(Show Context)
Citation Context ...haus hoped to linearize the problem by identifying all simple groups as groups of automorphisms of some linear structure, perhaps a finite Lie algebra. This approach gained credibility when Chevalley =-=[Ch]-=- found a uniform method to construct finite analogues of the simple complex Lie groups. Supplemented with variations by Steinberg [St] and Ree [Re1], [Re2], this furnished a Lie-theoretic context for ... |

36 |
The local structure of finite groups of characteristic 2 type
- Gorenstein, Lyons
- 1983
(Show Context)
Citation Context ...e 255 pages in length, and there was no fat on the manuscript. Many of the later manuscripts would follow this triple digit pattern, culminating in the 731 page Memoirs volume by Gorenstein and Lyons =-=[GL1]-=- and the forthcoming 800+ page Quasithin Paper by Aschbacher and S. D. Smith [AS]. Feit and Thompson published the Odd Order Paper in 1963 [FT]: The Odd Order Theorem. All finite groups of odd order a... |

35 |
The friendly giant
- Griess
- 1982
(Show Context)
Citation Context ...ng the existence and uniqueness of some of the sporadic simple groups, notably the Monster. However, in January 1980 Griess startled everyone by announcing a computer-free construction of the Monster =-=[Gr]-=-. Over the previous decade Sims had developed powerful methods for the computer construction of finite simple groups as permutation groups, achieving striking successes beginning with the construction... |

34 |
Theory and applications of finite groups
- Miller, Blichfeldt, et al.
- 1961
(Show Context)
Citation Context ... the main unsolved problem.” (“Als ungelöstes Hauptproblem verbleibt die Bestimmung aller endlichen einfachen Gruppen.”) In contrast to earlier texts by Burnside [Bu5], Miller, Blichfeldt and Dickson =-=[MBD]-=- (1916) and Speiser [Sp] (1927), Zassenhaus’ work almost completely omits the theories of permutation groups and of group characters, focussing singlemindedly on the architectural structure of groups ... |

31 |
Traité des substitutions et des équations algébiques
- Jordan
- 1870
(Show Context)
Citation Context ...ere some important pre-natal events. Galois introduced the concept of a normal subgroup in 1832, and Camille Jordan in the preface to his Traité des substitutions et des équations algebriques in 1870 =-=[J1]-=- flagged Galois’ distinction between groupes simples and groupes composées as the most important dichotomy in the theory of permutation groups. Moreover, in the Traité, Jordan began to build a databas... |

21 |
Theory of Groups of Finite Order, second edition
- Burnside
- 1911
(Show Context)
Citation Context ..., |G| must have at least seven prime factors, and G can have no proper subgroup of indexsTHE CLASSIFICATION OF THE FINITE SIMPLE GROUPS 317 less than 101. This prompted his famous observation in 1911 =-=[Bu5]-=-: “The contrast that these results shew between groups of odd and even order suggests inevitably that [nonabelian] simple groups of odd order do not exist.” The less-than-friendly rivalry between Frob... |

21 |
Central elements in core-free groups
- Glauberman
- 1966
(Show Context)
Citation Context ...al, wreathed, homocyclic abelian) or be of one exceptional isomorphism type. Alperin’s proof was made possible by an elegant application of modular character theory to 2-fusion analysis by Glauberman =-=[Gl1]-=-: Glauberman’s Z ∗ -Theorem. Let G be a finite group with no nontrivial normal subgroup of odd order. Let z be an involution of G. Eitherz ∈ Z(G) or z commutes with a G-conjugate z g with z g �= z. Al... |

19 | Sylow intersections and fusion
- Alperin
- 1967
(Show Context)
Citation Context ...rly embedded weakly closed subgroups of P .The best result in this vein was achieved in the 1970’s by Yoshida [Y]. The most general theorem, again of later vintage, is Alperin’s Fusion Theorem (1967) =-=[Al]-=-, which implies that the existence of abelian p-quotients is always determined p-locally, i.e. by examination of the full set of p-local data for G. Indeed it provides much sharper information. Also i... |

17 |
Transitive Gruppen gerader Ordnung, in denen jede Involution genau einen Punkt festläßt
- Bender
- 1971
(Show Context)
Citation Context ...t.)s332 RONALD SOLOMON One obvious gap between this definition and Suzuki’s hypotheses is that G is not assumed to be 2-transitive. Indeed this need not be the case if G has 2-rank 1. However, Bender =-=[Be3]-=- succeeded in 1968 in proving the following beautiful theorem: Bender’s Strongly Embedded Theorem. Let G be a finite simple group with a strongly embedded subgroup M. ThenG satisfies the hypotheses of... |

17 |
A note on soluble groups
- Hall
- 1928
(Show Context)
Citation Context ...fort to avoid having to study any p-groups beyond those masterfully analyzed by Philip Hall. Even more important for the future theory of simple groups was the series of papers Hall published in 1928 =-=[H1]-=- and 1937 [H3], [H4] on finite solvable groups. In the first he establishes generalizations of Sylow’s Theorems for finite solvable groups; namely if G is a finite solvable group of order mn with m an... |

17 |
A contribution to the theory of groups of prime power order
- Hall
- 1936
(Show Context)
Citation Context ...e suggestions as to worthwhile problems. This was in 1927 and shortly afterwards Burnside died. I never met him, but he has been the greatest influence on my ways of thinking.” Hall undertook in 1932 =-=[H2]-=- “the first stages of an attempt to construct a systematic general theory of groups of prime-power order,” justifying this project with the sentence: “It is widely recognised, I believe, that the asto... |

16 |
A characterization of Chevalley groups over fields of odd order
- Aschbacher
- 1975
(Show Context)
Citation Context ...ory Year at Rutgers University. (See [So] for a survey article.) A major ingredient in the proof is Aschbacher’s characterization of the groups of Lie types336 RONALD SOLOMON over fields of odd order =-=[A4]-=-, for which he was awarded the Cole Prize in Algebra. At this point the problem of groups of odd type was essentially “busted”, although a substantial amount of mopping-up in terms of specific standar... |

15 |
The characterization of finite groups with dihedral Sylow 2-subgroups
- Gorenstein, Walter
- 1965
(Show Context)
Citation Context ...rlier) shows that no simple group of even order except C2 has cyclic Sylow 2-subgroups. Brauer and Suzuki [BS] had proved that no simple group has quaternion Sylow 2-subgroups.) Gorenstein and Walter =-=[GW1]-=- completed the dihedral case. Luckily, an elegant argument of Alperin showed that a 2-group of 2-rank 2 which was a candidate to be a Sylow 2-subgroup of a simple group must fall into one of four infi... |

10 |
A characteristic subgroup of a p-stable group
- Glauberman
- 1968
(Show Context)
Citation Context ..., but the analysis of failure of factorization modules remains meaningful and is at the heart of the classification of simple groups of characteristic 2-type. Glauberman’s discovery of the ZJ-Theorem =-=[Gl2]-=- around 1967 provided an easier approach than factorization theorems in the context of groups of odd order. Recently Stellmacher [Sl1] has established an analogue of the ZJ-Theorem for groups of order... |

10 |
Recherches sur les substitutions
- Jordan
(Show Context)
Citation Context ... in the 1930’s was Zassenhaus’ extension of the work of Jordan and Frobenius on transitive permutation groups. Jordan had classified all sharply k-transitive permutation groups for k = 2 and for k ≥ 4=-=[J2]-=-. Sharply 2-transitive permutation groups of a finite set Ω satisfy the hypotheses of Frobenius’ Theorem and thus have the structure G = KGα where K is a regular normal subgroup of G. As K −{1} is per... |

9 |
On groups of even order
- Brauer, Fowler
- 1955
(Show Context)
Citation Context ... even order, given the order of one of its involution centralizers, and a characterization of PSL(3,q)andM11 via the centralizer of an involution. Because of its generality, the Brauer-Fowler Theorem =-=[BF]-=- had a particularly great psychological impact. Brauer-Fowler Theorem. Let G be a finite simple group of even order containing an involution t. If|CG(t)| = c, then|G| ≤(c 2 )!. The specific bound is u... |

8 |
The Classification of the Quasithin Groups
- Aschbacher, Smith
- 2004
(Show Context)
Citation Context ...anuscripts would follow this triple digit pattern, culminating in the 731 page Memoirs volume by Gorenstein and Lyons [GL1] and the forthcoming 800+ page Quasithin Paper by Aschbacher and S. D. Smith =-=[AS]-=-. Feit and Thompson published the Odd Order Paper in 1963 [FT]: The Odd Order Theorem. All finite groups of odd order are solvable. This short sentence and its long proof were a moment in the evolutio... |

8 |
Groups with (B; N)-pair of rank 2
- Fong, Seitz
- 1973
(Show Context)
Citation Context ...her cyclic nor dihedral). In particular all finite simple BN-pairs of rank at least 3 are split, in the sense that B = U(B ∩ N) whereU is a nilpotent normal subgroup of B and U ∩ N =1. Fong and Seitz =-=[FS]-=- extended Tits’ result by classifying split BN-pairs of rank 2. In particular every simple group G of Lie-type defined over a finite field of characteristic p is a split BN-pair with U aSylowp-subgrou... |

7 |
Local analysis for the Odd Order Theorem
- Bender, Glauberman
- 1994
(Show Context)
Citation Context ...ument of the Odd Order Paper and, using the earlier work of Dade and Sibley, obtained a revised proof [Pe2] of complementary material to the Bender-Glauberman Local Analysis for the Odd Order Theorem =-=[BG2]-=-. Peterfalvi also obtained a revised proof of Suzuki’s Theorem on 2transitive groups, as well as O’Nan’s characterization of PSU(3,q)forq odd [Pe1]. With Enguehard’s treatment [E] of the work of Thomp... |

7 |
On finite groups of even order whose 2-Sylow group is a quaternion group
- Brauer, Suzuki
- 1959
(Show Context)
Citation Context ...ate problem. (A 2-group of rank 1 is either cyclic or quaternion. An old argument (sketched earlier) shows that no simple group of even order except C2 has cyclic Sylow 2-subgroups. Brauer and Suzuki =-=[BS]-=- had proved that no simple group has quaternion Sylow 2-subgroups.) Gorenstein and Walter [GW1] completed the dihedral case. Luckily, an elegant argument of Alperin showed that a 2-group of 2-rank 2 w... |

7 |
On some properties of groups of odd order
- Burnside
- 1901
(Show Context)
Citation Context ...y primes p for which |PSL(2,p)| is a product of exactly six prime factors. Thus again this strategy rapidly encounters difficult and irrelevant obstacles. Its last hurrah was Burnside’s proof in 1900 =-=[Bu3]-=- that if G is a nonabelian simple group of odd order, then |G| must be a product of at least seven prime numbers. In a similar but more fruitful direction, Frobenius [Fr1] proved in 1893 that a simple... |

7 |
On the p-length of p-soluble groups and reduction theorems for Burnside's problem
- HALL, HIGMAN
(Show Context)
Citation Context ...e and Restricted Burnside Problems, including Kostrikin’s proof of the Restricted Burnside Theorem for groups of prime exponent. In this context Philip Hall and Graham Higman wrote a remarkable paper =-=[HH]-=- in 1956 aimed at the reduction of the Restricted Burnside Problem to the prime-power exponent case. They provided an insightful analysis of the structure of finite p-solvable groups (a class containi... |

6 |
Finite simple groups of 2-rank two
- Alperin, Brauer, et al.
- 1973
(Show Context)
Citation Context ..., which turned out to be quite difficult and to require considerable 2-modular character theory, for which they turned to Brauer. By 1969 simple groups of 2-rank at most 2 had been classified [ABG1], =-=[ABG2]-=-. Once it could be assumed that G had 2-rank at least 3, it was at least possible to begin 2-signalizer functor analysis. This was not, however, quite enough to pass gracefully from the conclusion of ... |

6 |
Finite groups whose 2-subgroups are generated by at most 4 elements
- Gorenstein, Harada
- 1974
(Show Context)
Citation Context ...clusion of the Signalizer Functor Theorem for p = 2 to the existence of a 2-Uniqueness Subgroup of the desired type when θ(G) �= 1. To bridge this gap Gorenstein and Harada produced a monumental work =-=[GH]-=- classifying simple groups of sectional 2-rank at most 4. Later Harada [Hr1] discovered a short and elegant argument to build the same bridge, though their magnum opus was quoted in many other context... |

5 |
Finite groups with quasidihedral and wreathed Sylow 2-subgroups
- Alperin, Brauer, et al.
- 1970
(Show Context)
Citation Context ...hed case, which turned out to be quite difficult and to require considerable 2-modular character theory, for which they turned to Brauer. By 1969 simple groups of 2-rank at most 2 had been classified =-=[ABG1]-=-, [ABG2]. Once it could be assumed that G had 2-rank at least 3, it was at least possible to begin 2-signalizer functor analysis. This was not, however, quite enough to pass gracefully from the conclu... |

5 |
On finite groups of component type
- Aschbacher
- 1975
(Show Context)
Citation Context ...tion-centralizer theorems would have to be proved. A first version of such a bounding theorem was proved by Powell and Thwaites [PT]. Shortly thereafter, an optimal theorem was obtained by Aschbacher =-=[A2]-=-,[A3] in 1973. Aschbacher’s Component Theorem. Let G be a finite simple group and suppose that the B-Theorem holds. Suppose that E(CG(t)) �= 1for some involution t of G. Then there exists an involutio... |

5 |
A characteristic property of soluble groups
- Hall
- 1937
(Show Context)
Citation Context ...having to study any p-groups beyond those masterfully analyzed by Philip Hall. Even more important for the future theory of simple groups was the series of papers Hall published in 1928 [H1] and 1937 =-=[H3]-=-, [H4] on finite solvable groups. In the first he establishes generalizations of Sylow’s Theorems for finite solvable groups; namely if G is a finite solvable group of order mn with m and n coprime, t... |

5 |
Theorems like Sylow’s
- Hall
- 1956
(Show Context)
Citation Context ...s, I shall discuss them along with related generalizations in the remainder of this section. In their attack on the “large” case, Feit and Thompson were motivated by an important paper of Philip Hall =-=[H5]-=- from 1956, extending his earlier work on criteria for solvability in terms of permutability of Sylow subgroups. Hall introduced the symbol Epq to denote the existence in G of a Hall {p, q}-subgroup. ... |

4 |
Thin finite simple groups
- Aschbacher
- 1978
(Show Context)
Citation Context ...fields of characteristic 2, but there are also a few sporadic examples, such as Co2, J4 and F3. With minor modification, Gorenstein’s eight steps were in fact implemented: IX: Thin Groups (Aschbacher =-=[A5]-=-); X: Groups with a Strongly p-Embedded (2-Local) Subgroup, p odd (Aschbacher [A8]); XI and XV: The Signalizer Functor Method and Component Theorem for Odd Primes (Gorenstein-Lyons [GL1]); XII: Groups... |

4 |
Simple Groups as Far as Order 660
- Cole
(Show Context)
Citation Context ...r threw down a gauntlet which was rapidly taken up by Frank Cole, who in 1892 [Co1] determined all simple groups of orders up to 500 (except for some uncertainties related to 360 and 432) and in 1893 =-=[Co2]-=- extended this up to 660, discovering in the process a new simple group SL(2, 8). By the dawn of the 20th century Miller and Ling (1900) [ML] had pushed this frontier out to 2001. These results were a... |

4 |
2-signalizer functors on finite groups
- Goldschmidt
- 1972
(Show Context)
Citation Context ...-signalizer functor. Then Iθ(A) has a unique maximal element θ(G). Early versions of this theorem were proved around 1969 by Gorenstein [G2]. The full theorem for p = 2 was established by Goldschmidt =-=[Go3]-=-, and then finally the general case was proved by Glauberman [Gl4] in 1973. The proof entails remaining in the realm of solvable θ-subgroups of G where it is possible to recover some of Thompson’s res... |

4 |
A uniqueness proof for the Monster
- Griess, Meierfrankenfeld, et al.
- 1989
(Show Context)
Citation Context ...e, Ly, ON, HN, Th, BM and M. (As uniqueness theorems trickled in, statement (d) became precisely accurate. For example, Griess, Meierfrankenfeld and Segev established the uniqueness of the Monster in =-=[GMS]-=-.) In an unpublished draft of the preface to his Finite Simple Groups: An Introduction to Their Classification [G4], Gorenstein asserts: “In August, 1980, the classification of the finite simple group... |

3 |
Finite groups with a proper 2-generated core
- Aschbacher
- 1974
(Show Context)
Citation Context ...y embedded subgroup M. ThenG satisfies the hypotheses of Suzuki’s Theorem. Thus G ∼ = SL(2, 2 n ), Sz(2 n ) or PSU(3, 2 n ). In 1973 building on work of Gorenstein and Walter and of Shult, Aschbacher =-=[A1]-=- was able to strengthen Bender’s Theorem to the precise 2-Uniqueness Theorem needed for the Signalizer Functor Method. Paralleling these developments was another line of research which yielded dramati... |

3 |
Über endliche Gruppen mit einer zu L2(2 n ) isomorphen Faktorgruppe
- Baumann
- 1979
(Show Context)
Citation Context ... failure-offactorization situations. A fundamental example occurs when N/O2(N) ∼ = PSL(2, 2 n )andV/Z(N) is a natural module for N/O2(N), i.e. a 2n-dimensional irreducible F2[N/O2(N)]-module. Baumann =-=[Ba]-=- analyzed this situation and was able to determine the possible structures for O2(N), introducing the Baumann subgroup which is crucial in circumventing the difficulty mentioned above. Glauberman and ... |

3 |
Theory of linear groups in an arbitrary field
- Dickson
- 1901
(Show Context)
Citation Context ...D1]. Dickson was well aware of the analogy between his work and the recent monumental results of Killing and Cartan classifying the simple continuous groups of Lie, and shortly thereafter (1901,1903) =-=[D2]-=-, [D3] he succeeded in constructing analogues of the Lie group G2 over finite fields F and establishing their simplicity when |F | > 2. He also studied finite analogues of E6 but did not prove their s... |

3 |
auflösbare Gruppen IV
- Frobenius
- 1901
(Show Context)
Citation Context ...ere exists a unique g ∈ G with g(x) =y. Every group acts regularly on itself via the regular representation. Thus regular action puts no restriction on the structure of G. On the other hand Frobenius =-=[Fr2]-=- proved the following result. Frobenius’ Theorem. Let G be a permutation group which acts transitively but not regularly on a finite set X. Suppose that no nonidentity permutation in G fixes more than... |

3 |
On solvable signalizer functors in finite groups
- Glauberman
- 1976
(Show Context)
Citation Context ... Early versions of this theorem were proved around 1969 by Gorenstein [G2]. The full theorem for p = 2 was established by Goldschmidt [Go3], and then finally the general case was proved by Glauberman =-=[Gl4]-=- in 1973. The proof entails remaining in the realm of solvable θ-subgroups of G where it is possible to recover some of Thompson’s results, such as the Transitivity Theorem. A Nonsolvable Signalizer F... |

2 |
Finite groups of rank 3
- Aschbacher
- 1981
(Show Context)
Citation Context ...rimes (Gorenstein-Lyons [GL1]); XII: Groups of Characteristic { 2,p }-Type (Timmesfeld et al. [Tm2]); XIII: Quasithin Groups (Mason [Ms2]; Aschbacher-Smith [AS]); XIV Groups with e(G) = 3 (Aschbacher =-=[A7]-=-); XVI Final Characterization of the Simple Groups of Characteristic 2-Type (Gilman-Griess [GG]). I have taken some small liberties with the names of Gorenstein’s steps.sTHE CLASSIFICATION OF THE FINI... |

2 |
The uniqueness case for finite groups
- Aschbacher
- 1983
(Show Context)
Citation Context ...2, J4 and F3. With minor modification, Gorenstein’s eight steps were in fact implemented: IX: Thin Groups (Aschbacher [A5]); X: Groups with a Strongly p-Embedded (2-Local) Subgroup, p odd (Aschbacher =-=[A8]-=-); XI and XV: The Signalizer Functor Method and Component Theorem for Odd Primes (Gorenstein-Lyons [GL1]); XII: Groups of Characteristic { 2,p }-Type (Timmesfeld et al. [Tm2]); XIII: Quasithin Groups ... |

2 |
On the uniqueness theorem
- Bender
- 1970
(Show Context)
Citation Context ...n abelian psubgroup of K, asp ranges over all prime divisors of |K|.) A beautiful proof of the Uniqueness Theorem, incorporating Glauberman’s ZJTheorem and his own new ideas, was discovered by Bender =-=[Be1]-=- in the late 1960’s. Around the same time, Gorenstein and Walter undertook a profound analysis of the Feit-Thompson signalizer arguments with the goal of extending the analysis toacontextwherep-local ... |

2 |
Characters of finite groups with dihedral Sylow 2-subgroups
- Bender, Glauberman
- 1981
(Show Context)
Citation Context ...se of Bender’s reluctance to use modular character theory. Finally Glauberman showed Bender how to replace the modular character theory with ordinary character theory and the work was published [BG1],=-=[Be4]-=-. Meanwhile in the late 1970’s Glauberman and Bender had been thinking independently about further revision of the local analysis of the Odd Order Paper, and Glauberman brought a completed manuscript ... |

2 |
Entwicklungslinien in der Theorie endlicher Gruppen
- Bender
- 1990
(Show Context)
Citation Context ... sein.” [“The most interesting problem seems to me to be the classification of all finite simple groups.”] So spoke Zassenhaus in his Antrittsvorlesung at the University of Hamburg in 1947 (quoted in =-=[Be5]-=-). The interest in the problem was clear. The best strategy for its solution was not at all clear. Inspired by the work of Killing and E. Cartan, Zassenhaus hoped to linearize the problem by identifyi... |

2 |
Simple groups from order 201 to order 500
- Cole
- 1892
(Show Context)
Citation Context ...low published his famous theorems on subgroups of prime power order [Sy]. Nevertheless Hölder’s paper is a landmark. Hölder threw down a gauntlet which was rapidly taken up by Frank Cole, who in 1892 =-=[Co1]-=- determined all simple groups of orders up to 500 (except for some uncertainties related to 360 and 432) and in 1893 [Co2] extended this up to 660, discovering in the process a new simple group SL(2, ... |

2 |
On groups whose orders are products of three prime factors
- Cole, Glover
(Show Context)
Citation Context ... idea that the structure of a finite group G depends more on the shape of the prime factorization of |G| than on the actual nature of the prime factors was explicitly stated by Cole and Glover (1893) =-=[CG]-=- in their critique of Cayley and Kempe: “It is however a defect of their method of classification that it proceeds simply according to the order and not the type of the groups. Thus the groups of orde... |

2 |
A new system of simple groups
- Dickson
- 1905
(Show Context)
Citation Context ...ickson was well aware of the analogy between his work and the recent monumental results of Killing and Cartan classifying the simple continuous groups of Lie, and shortly thereafter (1901,1903) [D2], =-=[D3]-=- he succeeded in constructing analogues of the Lie group G2 over finite fields F and establishing their simplicity when |F | > 2. He also studied finite analogues of E6 but did not prove their simplic... |