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A brief history of the classification of the finite simple groups

Author(s): Ronald Solomon
Journal: Bull. Amer. Math. Soc. 38 (2001), 315-352.
MSC (2000): Primary 20D05
Posted: March 27, 2001
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Abstract:

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

References:

[Al]
J. L. Alperin, Sylow intersections and fusion, J. Algebra 6 (1967), 222-241. MR 35:6748

[ABG1]
J. L. Alperin, R. Brauer and D. Gorenstein, Finite groups with quasi-dihedral and wreathed Sylow $2$-subgroups, Trans. Amer. Math. Soc. 151 (1970), 1-261. MR 44:1724

[ABG2]
J. L. Alperin, R. Brauer and D. Gorenstein, Finite simple groups of $2$-rank two, Scripta Math. 29 (1973), 191-214. MR 53:5728

[A1]
M. Aschbacher, Finite groups with a proper $2$-generated core, Trans. Amer. Math. Soc 197 (1974), 87-112. MR 51:681

[A2]
M. Aschbacher, On finite groups of component type, Illinois J. Math. 19 (1975), 78-115. MR 51:13018

[A3]
M. Aschbacher, Tightly embedded subgroups of finite groups, J. Algebra 42 (1976), 85-101. MR 54:10390

[A4]
M. Aschbacher, A characterization of Chevalley groups over fields of odd order. I, II, Ann. of Math. 106 (1977), 353-468. MR 58:16865a; MR 58:16865b

[A5]
M. Aschbacher, Thin finite simple groups, J. Algebra 54 (1978), 50-152. MR 82j:20032

[A6]
M. Aschbacher, A factorization theorem for $2$-constrained groups, Proc. London Math. Soc. (3) 43 (1981), 450-477. MR 83a:20019b

[A7]
M. Aschbacher, Finite groups of rank $3$, I, Invent. Math. 63 (1981), 357-402; II, Invent. Math. 71 (1983), 51-163. MR 82j:20033; MR 84h:20008

[A8]
M. Aschbacher, The uniqueness case for finite groups. I, II, Ann. of Math. 117 (1983), 383-551. MR 84g:20021a; MR 84g:20021b

[AS]
M. Aschbacher and S. D. Smith, A classification of quasithin groups, Amer. Math. Soc. Surveys and Monographs, to appear.

[Ba]
B. Baumann, Über endliche Gruppen mit einer zu $L_{2}(2^{n})$ isomorphen Faktorgruppe, Proc. Amer. Math. Soc. 74 (1979), 215-222. MR 80g:20024

[Be1]
H. Bender, On the uniqueness theorem, Illinois J. Math. 14 (1970), 376-384. MR 41:6959

[Be2]
H. Bender, On groups with abelian Sylow $2$-subgroups, Math. Z. 117 (1970), 164-176. MR 44:5378

[Be3]
H. Bender, Transitive Gruppen gerader Ordnung in denen jede Involution genau einen Punkt festlasst, J. Algebra 17 (1971), 527-554. MR 44:5370

[Be4]
H. Bender, Finite groups with dihedral Sylow $2$-subgroups, J. Algebra 70 (1981), 216-228. MR 83c:20011b

[Be5]
H. Bender, Entwicklungslinien in der Theorie endlicher Gruppen, Jber. d. Dt. Math.-Verein. (1992), 77-123.

[BG1]
H. Bender and G. Glauberman, Characters of finite groups with dihedral Sylow $2$-subgroups, J. Algebra 70 (1981), 200-215. MR 83c:20011a

[BG2]
H. Bender and G. Glauberman, Local Analysis for the Odd Order Theorem, London Math. Soc. Lecture Notes Series 188, Cambridge U. Press, Cambridge, 1994. MR 96h:20036

[Br1]
R. Brauer, On groups whose order contains a prime to the first power. I, II, Amer. J. Math. 64 (1942), 401-440. MR 4:1e; MR 4:2a

[Br2]
R. Brauer, Blocks of characters and structure of finite groups, Bull. Amer. Math. Soc. (New Series) 1 (1979), 21-38. MR 80f:20013

[BF]
R. Brauer and K. A. Fowler, On groups of even order, Ann. of Math. 62 (1955), 565-583. MR 17:580e

[BS]
R. Brauer and M. Suzuki, On finite groups of even order whose $2$-Sylow subgroup is a quaternion group, Proc. Natl. Acad. Sci. USA 45 (1959), 1757-1759. MR 22:731

[BSW]
R. Brauer, M. Suzuki and G. E. Wall, A characterization of the one-dimensional unimodular groups over finite fields, Illinois J. Math. 2 (1958), 718-745. MR 21:3487

[Bu1]
W. Burnside, Notes on the theory of groups of finite order, Proc. London Math. Soc. 26 (1895), 191-214.

[Bu2]
W. Burnside, On a class of groups of finite order, Trans. Cambridge Phil. Soc. 18 (1899), 269-276.

[Bu3]
W. Burnside, On some properties of groups of odd order, Proc. London Math. Soc. 32 (1900), 162-185, 257-268.

[Bu4]
W. Burnside, On groups of order $p^{\alpha }q^{\beta }$, Proc. London Math. Soc. (2) 1 (1904), 388-392.

[Bu5]
W. Burnside, Theory of Groups of Finite Order, Second edition, Cambridge University Press, Cambridge, 1911.

[Ch]
C. Chevalley, Sur certains groupes simples, Tôhoku Math. J. 7 (1955), 14-66. MR 17:457c

[Cl]
M. J. Collins (ed.), Finite Simple Groups II, Academic Press, London, 1980. MR 82h:20021

[Co1]
F. N. Cole, Simple groups from order $201$ to order $500$, Amer. J. Math. 14 (1892), 378-388.

[Co2]
F. N. Cole, Simple groups as far as order $660$, Amer. J. Math. 15 (1893), 303-315.

[CG]
F. N. Cole and J. W. Glover, On groups whose orders are products of three prime factors, Amer. J. Math 15 (1893), 191-220.

[Da]
E. Dade, Lifting group characters, Ann. of Math. 79 (1964), 590-596. MR 28:4023

[DGS]
A. Delgado, D. Goldschmidt and B. Stellmacher, Groups and Graphs: New Results and Methods, DMW Seminar Band 6, Birkhäuser Verlag, Basel, 1985. MR 88a:05076

[D1]
L. E. Dickson, Linear Groups with an Exposition of the Galois Field Theory, Teubner, Leipzig, 1901.

[D2]
L. E. Dickson, Theory of linear groups in an arbitrary field, Trans. Amer. Math. Soc. 2 (1901), 363-394.

[D3]
L. E. Dickson, A new system of simple groups, Math. Annalen 60 (1905), 137-150.

[Du]
P. Duren (ed.), A Century of Mathematics in America, Part I, American Mathematical Society, Providence, RI, 1988. MR 90a:01064

[E]
M. Enguehard, Obstructions et $p$-groupes de classe $3$. Caracterisation des groupes de Ree, Astérisque 142-143 (1986), 3-139. MR 88a:20028

[Fe]
W. Feit, Richard D. Brauer, Bull. Amer. Math. Soc. (New Series) 1 (1979), 1-20. MR 80m:01033

[FHT]
W. Feit, M. Hall, Jr. and J. G. Thompson, Finite groups in which the centralizer of any non-identity element is nilpotent, Math. Zeit. 74 (1960), 1-17. MR 22:5674

[FT]
W. Feit and J. G. Thompson, Solvability of groups of odd order, Pacific J. Math. 13 (1963), 775-1029. MR 29:3538

[Fi]
B. Fischer, Groups generated by $3$-transpositions, Invent. Math. 13 (1971), 232-246. MR 45:3557

[F]
H. Fitting, Beiträge zur Theorie der Gruppen endlicher Ordnung, Jahr. Deutsch. Math. Ver. 48 (1938), 77-141.

[FS]
P. Fong and G. M. Seitz, Groups with a $(B,N)$-pair of rank $2$. I, II, Invent. Math. 21 (1973), 1-57; Invent. Math. 24 (1974), 191-239. MR 50:7335

[Fr1]
F. G. Frobenius, Über auflösbare Gruppen, Sitzungsber. Kön. Preuss. Akad. Wiss. Berlin (1893), 337-345.

[Fr2]
F. G. Frobenius, Über auflösbare Gruppen IV, Sitzungsber. Kön. Preuss. Akad. Wiss. Berlin (1901), 1216-1230.

[GG]
R. H. Gilman and R. L. Griess, Finite groups with standard components of Lie type over fields of characteristic two, J. Algebra 80 (1983), 383-516. MR 84g:20024

[Gl1]
G. Glauberman, Central elements in core-free groups, J. Algebra 4 (1966), 403-420. MR 34:2681

[Gl2]
G. Glauberman, A characteristic subgroup of a $p$-stable group, Canad. J. Math. 20 (1968), 1101-1135. MR 37:6365

[Gl3]
G. Glauberman, Prime-power factor groups of finite groups, I, Math. Z. 107 (1968), 159-172; II, Math. Z. 117 (1970), 46-56. MR 39:298; MR 45:3553

[Gl4]
G. Glauberman, On solvable signalizer functors in finite groups, Proc. London Math. Soc. (3) 33 (1976), 1-27. MR 54:5341

[GN]
G. Glauberman and R. Niles, A pair of characteristic subgroups for pushing-up finite groups, Proc. London Math. Soc. (3) 46 (1983), 411-453. MR 84g:20042

[Go1]
D. M. Goldschmidt, A group-theoretic proof of the $p^{a}q^{b}$-theorem for odd primes, Math. Z. 113 (1970), 373-375. MR 43:2085

[Go2]
D. M. Goldschmidt, Solvable signalizer functors on finite groups, J. Algebra 21 (1972), 137-148. MR 45:6913

[Go3]
D. M. Goldschmidt, $2$-signalizer functors on finite groups, J. Algebra 21 (1972), 321-340. MR 48:2257

[Go4]
D. M. Goldschmidt, $2$-fusion in finite groups, Ann. of Math. 99 (1974), 70-117. MR 49:407

[Go5]
D. M. Goldschmidt, Automorphisms of trivalent graphs, Ann. of Math. 111 (1980), 377-406. MR 82a:05052

[G1]
D. Gorenstein, Finite Groups, Harper and Row, New York, 1968. MR 38:229

[G2]
-, On the centralizers of involutions in finite groups, J. Algebra 11 (1969), 243-277. MR 39:1540

[G3]
-, The classification of finite simple groups, Bull. Amer. Math. Soc. (New Series) 1 (1979), 43-199. MR 80b:20015

[G4]
-, Finite Simple Groups: An Introduction to their Classification, Plenum Press, New York, 1982. MR 84j:20002

[G5]
-, The Classification of Finite Simple Groups, Plenum Press, New York, 1983. MR 86i:20024

[G6]
-, The classification of the finite simple groups, a personal journey: the early years, pp. 447-476 in [Du]. MR 90g:01031

[GH]
D. Gorenstein and K. Harada, Finite groups whose $2$-subgroups are generated by at most $4$ elements, Memoirs Amer. Math. Soc. 147 (1974). MR 51:3290

[GL1]
D. Gorenstein and R. Lyons, The local structure of finite groups of characteristic $2$ type, Memoirs Amer. Math. Soc. 276 (1983). MR 84g:20025

[GL2]
-, A local characterization of some sporadic groups, unpublished typescript.

[GLS1]
D. Gorenstein, R. Lyons and R. Solomon, The Classification of the Finite Simple Groups, Number 1, Amer. Math. Soc. Surveys and Monographs 40, #1 (1994). MR 95m:20014

[GLS2]
-, The Classification of the Finite Simple Groups, Number 3, Amer. Math. Soc. Surveys and Monographs 40, #3 (1998). MR 98j:20011

[GW1]
D. Gorenstein and J. H. Walter, The characterization of finite groups with dihedral Sylow $2$-subgroups, J. Algebra 2 (1965), 85-151, 218-270, 354-393. MR 31:1297a

[GW2]
-, The $\pi $-layer of a finite group, Illinois J. Math. 15 (1971), 555-565. MR 44:6812

[GW3]
-, Balance and generation in finite groups, J. Algebra 33 (1975), 224-287. MR 50:10051

[Gr]
R. L. Griess, The friendly giant, Invent. Math. 69 (1982), 1-102. MR 84m:20024

[GMS]
R. L. Griess, U. Meierfrankenfeld and Y. Segev, A uniqueness proof for the Monster, Ann. of Math. 130 (1989), 567-602. MR 90j:20030

[GR]
K. W. Gruenberg and J. E. Roseblade, Group Theory: Essays for Philip Hall, Academic Press, London, 1984. MR 85k:20004

[Gu]
O. Grün, Beiträge zur Gruppentheorie I, II, Crelle J. 174 (1936), 1-14; J. Reine Angew. Math. 186 (1945), pp. 165-169. MR 10:504c

[Ha]
M. Hall, Jr., The Theory of Groups, Macmillan, New York, 1959. MR 21:1996

[H1]
P. Hall, A note on soluble groups, J. London Math. Soc. 3 (1928), 98-105.

[H2]
P. Hall, A contribution to the theory of groups of prime power order, Proc. London Math. Soc. (2) 36 (1933), 29-95.

[H3]
P. Hall, A characteristic property of soluble groups, J. London Math. Soc. 12 (1937), 198-200.

[H4]
P. Hall, On the Sylow systems of a soluble group, Proc. London Math. Soc. (2) 43 (1937), 316-323.

[H5]
P. Hall, Theorems like Sylow's, Proc. London Math. Soc. (3) 6 (1956), 286-304. MR 17:1052a

[HH]
P. Hall and G. Higman, The $p$-length of a $p$-soluble group and reduction theorem for Burnside's problem, Proc. London Math. Soc. (3) 6 (1956), 1-42. MR 17:344b

[Hr1]
K. Harada, Groups with nonconnected Sylow $2$-subgroups revisited, J. Algebra 70 (1981), 339-349. MR 82i:20023

[Hr2]
K. Harada, Michio Suzuki, in Advanced Studies in Pure Mathematics: Groups and Combinatorics, Japan Math. Society, to appear.

[Ho]
O. Hölder, Die einfachen Gruppen in ersten und zweiten Hundert der Ordnungszahlen, Math. Annalen 40 (1892), 55-88.

[Ja]
Z. Janko, A new finite simple group with abelian $2$-Sylow subgroups and its characterization, J. Algebra 3 (1966), 147-186. MR 33:1359

[J1]
C. Jordan, Traité des Substitutions et des Équations Algébriques, Gauthier-Villars, Paris, 1870.

[J2]
C. Jordan, Recherches sur les substitutions, J. de Math. Pures et Appl. 17 (1872), 345-361.

[KM]
K. Klinger and G. Mason, Centralizers of $p$-subgroups in groups of characteristic $2$, $p$-type, J. Algebra 37 (1975), 362-375. MR 52:10873

[Ms1]
G. Mason, Quasithin groups, pp. 181-197 in [Cl]. MR 82h:20021

[Ms2]
G. Mason, Quasithin finite groups, unpublished typescript.

[Ma]
H. Matsuyama, Solvability of groups of order $2^{a}p^{b}$, Osaka J. Math. 10 (1973), 375-378. MR 48:2243

[McB1]
P. P. McBride, Near solvable signalizer functors on finite groups, J. Algebra 78 (1982), 181-214. MR 84g:20049a

[McB2]
P. P. McBride, Nonsolvable signalizer functors on finite groups, J. Algebra 78 (1982), 215-238. MR 84g:20049b

[MBD]
G. A. Miller, H. F. Blichfeldt and L. E. Dickson, Theory and Applications of Finite Groups, John Wiley and Sons, New York, 1916.

[ML]
G. A. Miller and G. H. Ling, Proof that there is no simple group whose order lies between $1092$and $2001$, Amer. J. Math. 22 (1900), 13-26.

[Pe1]
T. Peterfalvi, Sur la caractérisation des groupes $U_{3}(q)$, pour $q$ impair, J. Algebra 141 (1991), 253-264. MR 93c:20034

[Pe2]
T. Peterfalvi, Character Theory for the Odd Order Theorem, London Math. Soc. Lecture Note Series # 272, Cambridge U. Press, Cambridge, 2000. MR 2001a:20016

[PT]
M. Powell and G. Thwaites, On the nonexistence of certain types of subgroups in simple groups, Quart. J. Math. Oxford Series (2) 26 (1975), 243-256. MR 56:466

[Rd]
L. Rédei, Ein Satz über die endlichen einfachen Gruppen, Acta Math. 84 (1950), 129-153. MR 13:907d

[Re1]
R. Ree, A family of simple groups associated with the simple Lie algebra of type ($F_{4}$), Amer. J. Math. 83 (1961), 401-420. MR 24:A2617

[Re2]
R. Ree, A family of simple groups associated with the simple Lie algebra of type ($G_{2}$), Amer. J. Math. 83 (1961), 432-462. MR 25:2123

[Ro]
J. E. Roseblade, Obituary: Philip Hall, Bull. London Math. Soc. 16 (1984), 603-626. MR 86c:01056

[Se]
G. M. Seitz, Standard subgroups in finite groups, pp. 41-62 in [Cl]. MR 82h:20021

[So]
R. M. Solomon, The $B(G)$-conjecture and unbalanced groups, pp. 63-87 in [Cl]. MR 82h:20021

[Sp]
A. Speiser, Theorie der Gruppen von endlicher Ordnung, Springer-Verlag, Berlin, 1937. MR 82j:20003

[St]
R. Steinberg, Variations on a theme of Chevalley, Pacific J. Math. 9 (1959), 875-891. MR 22:79

[Sl1]
B. Stellmacher, A characteristic subgroup of $\Sigma _{4}$-free groups, Israel J. Math. 94 (1996), 367-379. MR 97d:20018

[Sl2]
B. Stellmacher, An application of the amalgam method: the $2$-local structure of $N$-groups of characteristic $2$-type, J. Algebra 190 (1997), 11-67. MR 98d:20018

[Sl3]
B. Stellmacher, Finite groups all of whose maximal parabolic subgroups are thin and of characteristic $2$-type, unpublished typescript.

[Su1]
M. Suzuki, On the finite group with a complete partition, J. Math. Soc. Japan 2 (1950), 165-185. MR 13:907b

[Su2]
M. Suzuki, The nonexistence of a certain type of simple groups of odd order, Proc. Amer. Math. Soc. 8 (1957), 686-695. MR 19:248f

[Su3]
M. Suzuki, A new type of simple groups of finite order, Proc. Nat. Acad. Sci. U.S.A. 46 (1960), 868-870. MR 22:11038

[Su4]
M. Suzuki, On a class of doubly transitive groups, Ann. of Math. 75 (1962), 105-145. MR 25:112

[Su5]
M. Suzuki, On a class of doubly transitive groups. II, Ann. of Math. 79 (1964), 514-589. MR 29:144

[Sy]
L. Sylow, Théorèmes sur les groupes de substitutions, Math. Ann. 5 (1872), 584-594.

[Sz]
G. Szekeres, Determination of a certain family of finite metabelian groups, Trans. Amer. Math. Soc. 66 (1949), 1-43. MR 11:320i

[T1]
J. G. Thompson, Normal $p$-complements for finite groups, Math. Z. 72 (1959/60), 332-354. MR 22:8070

[T2]
-, Nonsolvable finite groups all of whose local subgroups are solvable. I,II,III,IV,V,VI, Bull. Amer. Math. Soc. 74 (1968), 383-437 MR 37:6367; Pacific J. Math. 33 (1970), 451-536 MR 43:2072; Pacific J. Math. 39 (1971), 483-534 MR 47:1933; Pacific J. Math. 48 (1973), 511-592 MR 51:5745; Pacific J. Math. 50 (1974), 215-297; Pacific J. Math. 51 (1974), 573-630.

[T3]
-, Quadratic pairs, Actes du Congrès Intl. des Math. (Nice, 1970), Tome 1, Gauthier-Villars, Paris, 1971. MR 55:3051

[T4]
-, Finite non-solvable groups, pp. 1-12 in [GR]. MR 85k:20004

[Tm1]
F. G. Timmesfeld, Groups generated by root involutions. I, II, J. Algebra 33 (1975), 75-134; J. Algebra 35 (1975), 367-441. MR 51:8236

[Tm2]
F. G. Timmesfeld, Finite simple groups in which the generalized Fitting-group of the centralizer of some involution is extraspecial, Ann. of Math. 107 (1978), 297-369. MR 81i:20016a

[Ti]
J. Tits, Buildings of spherical type and finite BN-pairs, Lecture Notes in Mathematics # 386, Springer-Verlag, Berlin, 1974. MR 57:9866

[Wa]
J. H. Walter, The characterization of finite groups with abelian Sylow $2$-subgroups, Ann. of Math. 89 (1969), 405-514. MR 40:2749

[Wi]
H. Wielandt, $p$-Sylowgruppen und $p$-Faktorgruppen, J. Reine Angew. Math. 182 (1940), 180-193. MR 2:216e

[Y]
T. Yoshida, Character-theoretic transfer, J. Algebra 52 (1978), 1-38. MR 58:11095

[Z1]
H. J. Zassenhaus, Kennzeichnung endlicher linearer Gruppen, Abh. Math. Sem. Hamburg 11 (1936), 17-40.

[Z2]
H. J. Zassenhaus, Über endliche Fastkörper, Abh. Math. Sem. Hamburg 11 (1936), 187-220.

[Z3]
H. J. Zassenhaus, The Theory of Groups, Second Edition, Chelsea, New York, 1949. MR 11:77d

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Additional Information:

Ronald Solomon
Affiliation: Department of Mathematics, The Ohio State University, Columbus, OH 43210
Email: solomon@math.ohio-state.edu

DOI: 10.1090/S0273-0979-01-00909-0
PII: S 0273-0979(01)00909-0
Received by editor(s): September 18, 2000, and in revised form December 15, 2000
Posted: March 27, 2001
Additional Notes: Research partially supported by an NSF grant
Copyright of article: Copyright 2001, American Mathematical Society

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