Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
|
   
Available in electronic format
Available in print format
Bulletin of the American Mathematical Society
Bulletin of the American Mathematical Society
ISSN 1088-9485(e) ISSN 0273-0979(p)

     

Classifying the finite simple groups

Author(s): Daniel Gorenstein
Journal: Bull. Amer. Math. Soc. 14 (1986), 1-98.
MSC (1985): Primary 20D05, 20-02
MathSciNet review: 818060
Retrieve article in: PDF

References | Similar articles | Additional information

References:

1.
J. L. Alperin, Sylow intersections and fusion, J. Algebra 6 (1967), 222-241. MR 215913
2.
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 284499
3.
J. L. Alperin, R. Brauer, and D. Gorenstein, Finite simple groups of 2-rank two, Scripta Math. 29 (1973), 191-214. MR 401902
4.
M. Aschbacher, Finite groups with a proper 2-generated core, Trans. Amer. Math. Soc. 197 (1974), 87-223. MR 364427
5.
M. Aschbacher, Finite groups generated by odd transpositions. I, II, III, IV, Math Z. 127 (1972), 45-56; J. Algebra 26 (1973), 451-459; 26 (1973), 460-478; 26 (1973), 479-491. MR 310058
6.
M. Aschbacher, On finite groups of component type, Illinois J. Math. 19 (1975), 78-115. MR 376843
7.
M. Aschbacher, A characterization of Chevalley groups over fields of odd characteristic, Ann. of Math. (2) 106 (1977), 353-468. MR 498828
8.
M. Aschbacher, On finite groups in which the generalized Fitting group of the centralizer of some involution is symplectic but not extra-special, Comm. Algebra 4 (1976), 595-616. MR 407144
9.
M. Aschbacher, On finite groups in which the generalized Fitting group of the centralizer of some involution is extra-special, Illinois J. Math. 21 (1977), 347-364. MR 442089
10.
M. Aschbacher, A factorization theorem for 2-constrained groups, Proc. London Math. Soc. 43 (1981), 450-477. MR 635565
11.
M. Aschbacher, Standard components of alternating type centralized by a4-group (preprint). MR 401903
12.
M. Aschbacher, On the maximal subgroups of the finite classical groups, Invent. Math. 76 (1984), 469-514. MR 746539
13.
M. Aschbacher, Thin finite simple groups, J. Algebra 54 (1978), 50-152. MR 511458
14.
M. Aschbacher, Finite groups of rank 3. I, II, Invent. Math. 63 (1981), 357-402; 71 (1983), 51-162. MR 620676
15.
M. Aschbacher, Weak closure in finite groups of even characteristic, J. Algebra 70 (1981), 561-627. MR 623826
16.
M. Aschbacher, The uniqueness case for finite groups, Ann. of Math. (2)117 (1983), 383-551. MR 690850
17.
M. Aschbacher, Subgroup structure of finite groups, Proc. Rutgers Group Theory Year 1983-1984, Cambridge Univ. Press, New York-Cambridge, 1984, pp. 35-45. MR 817233
18.
M. Aschbacher, Finite groups, Cambridge Univ. Press, New York-Cambridge (to appear). MR 817233
19.
M. Aschbacher, Overgroups of Sylow subgroups in sporadic groups, Mem. Amer. Math. Soc. (to appear). MR 831891
20.
M. Aschbacher, D. Gorenstein, and R. Lyons, The embedding of 2-locals in finite groups of characteristic 2 type, Ann. of Math. (2)114 (1981), 335-455. MR 632843
21.
M. Aschbacher and L. Scott, Maximal subgroups of finite groups, J. Algebra 92 (1985). MR 772471
22.
M. Atiyah, An interview with Michael Atiyah, Math. Intelligencer 6 (1984), 9-19. MR 735359
23.
B. Baumann, Endliche Gruppen mit einer 2-zentralen Involution, deren Zentralisator 2-abgeschlossen ist, Illinois J. Math. 22 (1978), 240-261. MR 490825
24.
G. Belyĭ, On Galois extensions of a maximal cyclotomic field, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 267-276 = Math. USSR-Izv. 14 (1980), 247-256. MR 534593
25.
H. Bender, Transitive Gruppen gerader Ordnung, in denen jede Involution genau einen Punkt festlasst, J. Algebra 17 (1971), 527-554. MR 288172
26.
H. Bender, On groups with abelian Sylow 2-subgroups, Math. Z. 117 (1970), 164-176. MR 288180
27.
H. Bender, On the uniqueness theorem, Illinois J. Math. 14 (1970), 376-384. MR 262351
28.
H. Bender, Goldschmidt's 2-signalizer functor theorem, Israel J. Math. 22 (1975), 208-213. MR 390056
29.
H. Bender, Finite groups with dihedral Sylow 2-subgroups, J. Algebra 70 (1981), 216-228. MR 618389
30.
R. Blattner, Induced and produced representations of Lie algebras, Trans. Amer. Math. Soc. 144 (1969), 457-474. MR 308223
31.
R. Block and R. Wilson, The restricted simple Lie algebras are of classical or Cartan type, Proc. Nat. Acad. Sci. U.S.A. 81 (1984), 5271-5274. MR 758423
32.
E. Bombieri, Thompson's problem2 = 3), Invent. Math. 58 (1980), 77-100. MR 570875
33.
A. Borel and J. Tits, Elements unipotents et sousgroupes paraboliques de groupes reductifs. I, Invent. Math. 12 (1971), 97-104. MR 294349
34.
R. Brauer, On the structure of groups of finite order, Proc. Internat. Congr. Math., vol. 1, Noordhoff, Groningen, North-Holland, Amsterdam, 1954, pp. 209-217. MR 95203
35.
R. Brauer, Some applications of the theory of characters of finite groups. II. J. Algebra 1 (1964), 307-334. MR 174636
36.
R. Brauer and K. Fowler, On groups of even order, Ann. of Math. (2) 62 (1955), 565-583. MR 74414
37.
R. Brauer and M. Suzuki, On finite groups of even order whose 2-Sylow group is a quaternion group, Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 1757-1759. MR 109846
38.
F. Buekenhout, Diagrams for geometries and groups, J. Combinatorial Theory Ser. A 27 (1979), 121-151. MR 542524
39.
P. Cameron, S. Praeger, J. Saxl, and G. Seitz, The Sims conjecture (to appear).
40.
G. Cherlin, L. Harrington, and A. Lachlan, א (to appear). MR 779159
41.
G. Cherlin and A. Lachlan, Stable finitely homogeneous structures (to appear). MR 846608
42.
C. Chevalley, Sur certains groupes simples, Tôhoku Math. J. 7 (1955), 14-66. MR 73602
43.
J. Conway, A characterization of the Leech lattice, Invent. Math. 7 (1969), 137-142. MR 245518
44.
J. Conway and S. Norton, Monstrous moonshine, Bull. London Math. Soc. 11 (1979), 308-339. MR 554399
45.
C. Curtis, Central extensions of groups of Lie type, J. Reine Angew. Math. 220 (1965), 174-185. MR 188299
46.
A. Delgado and B. Stellmacher, Weak BN-pairs of rank2 (preprint).
47.
E. Dynkin, Maximal subgroups of classical groups, Trudy Moskov. Mat. Obshch. 1 (1952), 39-166 = Amer. Math. Soc. Transl. (2)6 (1957), 245-378. MR 49903
48.
E. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Mat. Sb. (N.S.) 30(72) (1952), 349-462 = Amer. Math. Soc. Transl. (2)6 (1957), 111-244. MR 47629
49.
F. Dyson, Unfashionable pursuits, Math. Intelligencer 5 (1983), 47-54. MR 737690
50.
M. Enguehard, Caractérisation des groupes de Ree (preprint). MR 873958
51.
B. Fein, W. Kantor, and M. Schacher, Relative Brauer groups. II, J. Reine Angew. Math. 328 (1981), 39-57. MR 636194
52.
W. Feit, Rigidity of PSL ≡ ±2 (mod 5), p ≠ 2, Proc. Rutgers Group Theory Year 1983-1984, Cambridge Univ. Press, New York-Cambridge, 1984, pp. 351-356. MR 817268
53.
W. Feit and P. Fong, Rational rigidity of G, Proc. Rutgers Group Theory Year 1983-1984, Cambridge Univ. Press, New York-Cambridge, 1984, pp. 323-326. MR 817266
54.
W. Feit, P. Fong, and B. Srinivasan, Rigidity of SL, Proc. Rutgers Group Theory Year 1983-1984, Cambridge Univ. Press, New York-Cambridge, 1984, pp. 303-308. MR 817263
55.
W. Feit and J. Thompson, Solvability of groups of odd order, Pacific J. Math. 13 (1963), 775-1029. MR 166261
56.
B. Fischer, Finite groups generated by 3-transpositions, Invent. Math. 13 (1971), 232-246. MR 294487
57.
P. Fong, Characters arising in the monster-modular connection, Proc. Sympos. Pure Math., vol. 37, Amer. Math. Soc., Providence, R. I., 1980, pp. 557-560. MR 604633
58.
R. Foote, Aschbacher blocks, Proc. Sympos. Pure Math., vol. 37, Amer. Math. Soc., Providence, R.I., 1980, pp. 37-42. MR 604554
59.
I. Frenkel, J. Lepowsky and A. Meurman, A natural representation of the Fischer-Griess monster with the modular function J as character, Proc. Nat. Acad. Sci. U.S.A. 81 (1984), 3256-3260. MR 747596
60.
R. Gilman, Components of finite groups, Comm. Algebra 4 (1976), 1133-1198. MR 430053
61.
R. Gilman and R. Griess, Finite groups with standard components of Lie type over fields of characteristic 2, J. Algebra 80 (1983), 383-516. MR 691810
62.
G. Glauberman, Central elements in core-free groups, J. Algebra 4 (1966), 403-420. MR 202822
63.
G. Glauberman, On solvable signalizer functors in finite groups, Proc. London Math. Soc. 33 (1976), 1-27. MR 417284
64.
G. Glauberman, Factorizations in local subgroups of finite groups, CBMS Regional Conf. Ser. in Math., no. 33, Amer. Math. Soc., Providence, R.I., 1977. MR 470072
65.
G. Glauberman, Local analysis in the odd order paper, Proc Sympos. Pure Math., vol. 37, Amer. Math. Soc., Providence, R.I., 1980, p. 137.
66.
G. Glauberman and R. Niles, A pair of characteristic subgroups for pushing up, Proc. London Math. Soc. (3) 46 (1983), 411-453. MR 699096
67.
D. Goldschmidt, Solvable signalizer functors on finite groups, J. Algebra 21 (1972), 137-148. MR 297861
68.
D. Goldschmidt, 2-signalizer functors on finite groups, J. Algebra 21 (1972), 321-340. MR 323904
69.
D. Goldschmidt, 2-fusion in finite groups, Ann. of Math. (2)99 (1974), 70-117. MR 335627
70.
D. Goldschmidt, Strongly closed 2-subgroups of finite groups, Ann. of Math. (2) 102 (1975), 475-489. MR 393223
71.
D. Goldschmidt, Automorphisms of trivalent graphs, Ann. of Math. (2) 111 (1980), 377-406. MR 569075
72.
D. Goldschmidt, Lectures on character theory, Publish or Perish Press, Berkeley, 1980. MR 575526
73.
D. Gorenstein, On the centralizers of involutions in finite groups, J. Algebra 11 (1969), 243-277. MR 240188
74.
D. Gorenstein, Finite groups, Harper and Row, New York, 1968; 2nd ed., Chelsea, New York, 1980. MR 231903
75.
D. Gorenstein, The classification of the finite simple groups, Bull. Amer. Math. Soc. 1 (1979), 43-199. MR 513750
76.
D. Gorenstein, Finite simple groups, Plenum, New York, 1982. MR 698782
77.
D. Gorenstein, The classification of finite simple groups, Plenum, New York, 1983. MR 746470
78.
D. Gorenstein and K. Harada, Finite groups whose 2-subgroups are generated by at most 4 elements, Mem. Amer. Math. Soc No. 147 (1974), 1-464. MR 367048
79.
D. Gorenstein and R. Lyons, Nonsolvable finite groups with solvable 2-local subgroups, J. Algebra 38 (1976), 453-522. MR 407128
80.
D. Gorenstein and R. Lyons, The local structure of finite groups of characteristic 2 type, Mem. Amer. Math. Soc. No. 42 (1983), 1-731. MR 690900
81.
D. Gorenstein and J. Walter, On the maximal subgroups of finite simple groups, J. Algebra 1 (1964), 168-231. MR 172917
82.
D. Gorenstein and J. Walter, The characterization of finite groups with dihedral Sylow 2-subgroups, J. Algebra 2 (1964), 85-151, 218-270, 354-393. MR 177032
83.
D. Gorenstein and J. Walter, Centralizers of involutions in balanced groups, J. Algebra 20 (1972), 284-319. MR 292927
84.
D. Gorenstein and J. Walter, Balance and generation in finite groups, J. Algebra 33 (1975), 224-287. MR 357583
85.
R. Griess, The friendly giant, Invent. Math. 69 (1982), 1-102. MR 671653
86.
G. Higman, Suzuki 2-groups, Illinois J. Math. 7 (1963), 79-96. MR 143815
87.
G. Hoyden, Realisierung der Jankogrouppen J1 and J, Algebra J. (to appear).
88.
G. Hoyden and B. Matzat, Realisierung sporadischer einfacher gruppen als Galoisgruppen ober Kreisteilungskorpern (to appear).
89.
D. Hunt, Rational rigidity and the sporadic groupsJ. Algebra (to appear). MR 837564
90.
M. Isaacs, Character theory of finite groups, Academic Press, New York, 1976. MR 460423
91.
Z. Janko, A new finite simple group of order86, 775, 571, 046, 077, 562, 880 which possesses M, J. Algebra 42 (1976), 564-496. MR 432751
92.
Z. Janko, Nonsolvable finite groups all of whose 2-local subgroups are solvable. I, J. Algebra 21 (1972), 458-517. MR 357584
93.
V. Kac., On the classification of simple Lie algebras over a field of nonzero characteristic, Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), 385-408; English transl. Math. USSR-Izv. 4 (1970), 391-413. MR 276286
94.
V. Kac., Description of filtered Lie algebras with which graded Lie algebras of Carton type are associated, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 800-834; English transl. Math. USSR-Izv. 8 (1974), 801-835. MR 369452
95.
W. Kantor, Polynomial-time algorithms for finding elements of prime order and Sylow subgroups (submitted).
96.
K. Klinger and G. Mason, Centralizers of p-groups in groups of characteristic 2, p-type, J. Algebra 37 (1975), 362-375. MR 390047
97.
A. Kostrikin and I. Shafarevich, Graded Lie algebras of finite characteristic, Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969), 748-764; English transl. Math. USSR-Izv. 3 (1969), 237-304. MR 252460
98.
J. Leech, Some sphere packings in higher space, Canad. J. Math. 16 (1964), 657-682. MR 167901
99.
J. Leech, Notes on sphere packings, Canad. J. Math. 19 (1967), 251-267. MR 209983
100.
E. Luks, Isomorphism of graphs of bounded valence can be tested in polynomial time, J. Comp. System. Sci. 25 (1982), 42-65. MR 685360
101.
J. Lundgren, On finite simple groups all of whose 2-local subgroups are solvable, J. Algebra 27 (1973), 491-515. MR 332953
102.
R. Lyons, A characterization of PSU (3, 4), Trans. Amer. Math. Soc. 164 (1972), 371-387. MR 286881
103.
A. MacWilliams, On 2-groups with no normal abelian subgroups of rank3 and their occurrence as Sylow 2-subgroups of finite simple groups, Trans. Amer. Math. Soc. 150 (1970), 345-408. MR 276324
104.
G. Mason, The classification of quasithin simple groups (preprint).
105.
G. Mason, Modular forms and the theory of Thompson series, Proc. Rutgers Group Theory Year 1983-1984, Cambridge Univ. Press, New York-Cambridge, 1984, 391-408. MR 817271
106.
B. Matzat, Zum einbettungsproblem der algebraischen zahlentheorie mit nicht abelschem kern (to appear) MR 788415
107.
B. Matzat and A. Zeh, Realisierung der mathieugruppen M. Number Theory (to appear).
108.
P. McBride, Nonsolvable signalizer functors on finite groups (preprint). MR 677718
109.
E. Noether, Gleichungen mit Vorgeschriebener Gruppe, Math. Ann. 78 (1918), 221-229. MR 1511893
110.
T. Peterfalvi, Le Théorème de Bender-Suzuki. I, II (preprints). MR 873959
111.
K. Phan, On groups generated by SU (3, q)'s. I, II, J. Austral. Math. Soc. 23 (1977), 66-76; 129-146. MR 447427
112.
R. Ree, A family of simple groups associated with the simple Lie algebra F4, Amer. J. Math. 83 (1961), 401-420. MR 132781
113.
R. Ree, A family of simple groups associated with the simple Lie algebra G, Amer. J. Math. 83 (1961), 432-462. MR 138680
114.
M. Ronan and S. Smith, 2-local geometries for some sporadic groups, Proc. Sympos. Pure Math., vol. 37, Amer. Math. Soc., Providence, R.I., 1980, pp. 283-289. MR 604595
115.
G. Seitz, The root subgroups for maximal tori in finite groups of Lie type, Pacific J. Math. 106 (1983), 153-244. MR 694680
116.
G. Seitz, Overgroups of irreducible linear groups, Proc. Rutgers Group Theory Year 1983-1984, Cambridge Univ. Press, New York-Cambridge, 1984, pp. 95-106. MR 817240
117.
I. Shafarevich, Construction of fields of algebraic numbers with given solvable Galois group, Izv. Akad. Nauk SSSR Ser. Mat. 18 (1954), 525-578. MR 71469
118.
D. Sibley, Groups of odd order revisited (in preparation).
119.
C. Sims, Graphs and finite permutation groups. I, II, Math. Z. 95 (1967), 76-86; 103 (1968), 276-281. MR 204509
120.
C. Sims, Some group theoretic algorithms, Lecture Notes in Math., vol. 697, Springer-Verlag, Berlin and New York, 1978, pp. 108-124. MR 524367
121.
F. Smith, Finite simple groups all of whose 2-local subgroups are solvable, J. Algebra 34 (1975), 481-520. MR 376847
122.
S. Smith, Large extraspecial subgroups of widths 4 and 6, J. Algebra 58 (1979), 251-280. MR 540638
123.
S. Smith, A characterization of orthogonal groups over GF(2), J. Algebra 62 (1980), 39-60. MR 561116
124.
S. Smith, A characterization of finite Chevalley and twisted groups of type E over GF(2), J. Algebra 62 (1980), 101-117. MR 561119
125.
S. Smith, On the head character of the monster simple group, Proc. Montreal Finite Group Theory Conference, 1982 (John McKay, ed.) (to appear). MR 822245
126.
R. Steinberg, Variations on a theme of Chevalley, Pacific J. Math. 9 (1959), 875-891. MR 109191
127.
R. Steinberg, Generateurs, relations, et revêtements de groupes algébriques, Colloque sur la theorie des groupes algébriques, C.B.R.M., Brussels, 1962, pp. 113-127. MR 153677
128.
R. Steinberg, Lectures on Chevalley groups, Lecture Notes, Yale Univ. 1967-1968. MR 466335
129.
R. Steinberg, Endomorphisms of linear algebraic groups, Mem. Amer. Math. Soc. No. 80 (1968). MR 230728
130.
G. Stroth, Solvable uniqueness subgroups (to appear).
131.
M. Suzuki, Finite groups with nilpotent centralizers, Trans. Amer. Math. Soc. 99 (1961), 425-470. MR 131459
132.
M. Suzuki, On a class of double transitive groups. I, II, Ann. of Math. (2) 75 (1962). MR 136646
133.
M. Suzuki, On characterizations of linear groups. III, Nagoya J. Math. 21 (1962), 159-183. MR 142665
134.
J. Thompson, Nonsolvable finite groups all of whose local subgroups are solvable. I-VI, Bull. Amer. Math. Soc. 74 (1968), 383-437; Pacific J. Math. 33 (1970), 451-536; 39 (1971), 483-534; 48 (1973), 511-592; 50 (1974), 215-297; 51 (1974), 573-630. MR 276325
135.
J. Thompson, Toward a characterization of E. I, II, III, J. Algebra 7 (1967), 406-414; 20 (1972), 610-621; 49 (1977), 163-166.
136.
J. Thompson, Some numerology between the Fischer-Griess monster and the elliptic modular function, Bull. London Math. Soc. 11 (1979), 352-353. MR 554402
137.
J. Thompson, Some finite groups which appear as ${\rm Gal}\,L/K$, where $K\subseteq Q(µ\sbn)$, J. Algebra 89 (1984), 437-499.
138.
J. Thompson, PSL, Proc. Rutgers Group Theory Year 1983-1984, Cambridge Univ. Press, New York-Cambridge, 1984, pp. 309-320. MR 1123999
139.
J. Thompson, Rational rigidity ofG2 (5), Proc. Rutgers Group Theory Year 1983-1984, Cambridge Univ. Press, New York-Cambridge, 1984, pp. 321-322. MR 817265
140.
J. Thompson, Primitive roots and rigidity, Proc. Rutgers Group Theory Year 1983-1984, Cambridge Univ. Press, New York-Cambridge, 1984, pp. 327-350. MR 817267
141.
F. Timmesfeld, Groups generated by root involutions. I, J. Algebra 33 (1975), 75-135. MR 372019
142.
F. Timmesfeld, Finite simple groups in which the generalized Fitting group of the centralizer of some involution is extra-special, Ann. of Math. (2)107 (1978), 297-369. MR 486255
143.
F. Timmesfeld, Tits geometries and parabolic systems in finite groups. I, II, Math. Z. 184 (1983), 377-396; 449-487.
144.
J. Tits, Algebraic and abstract simple groups, Ann. of Math. (2)80 (1964), 313-329. MR 164968
145.
J. Tits, Buildings of spherical type and finite (B, N)-pairs, Lecture Notes in Math., vol. 386, Springer-Verlag, Berlin and New York, 1974. MR 470099
146.
J. Tits, A local approach to buildings, The Geometric Vein, (Coxeter-Festschrift), Springer-Verlag, Berlin and New York, 1981, pp. 519-547. MR 661801
147.
J. Tits, Affine buildings of rank $\geqslant 4$ (unpublished).
148.
W. Tutte, A family of cubical graphs, Proc. Cambridge Philos. Soc. 43 (1947), 459-474. MR 21678
149.
W. Tutte, On the symmetry of cubic graphs, Canad. J. Math. 11 (1959), 621-624. MR 109794
150.
J. Walter, The characterization of finite groups with abelian Sylow 2-subgroups, Ann. of Math. (2) 89 (1969), 405-514. MR 249504
151.
J. Walter, Classical groups as Galois groups, Proc. Rutgers Group Theory Year 1983-1984, Cambridge Univ. Press, New York-Cambridge, 1984, pp. 385-390. MR 817269
152.
J. Walter, The B-conjecture; 2-components in finite simple groups (to appear). MR 604557
153.
W. Wong, Determination of a class of primitive permutation groups, Math. Z. 99 (1967), 235-246. MR 214653
154.
B. Zilber, Strongly minimal countably categorical theories, Siberian Math. J. 21 (1980), 98-112. (Russian) MR 569179
155.
T. Peterfalvi, Simplification du chapitre VI de l'article de Feit et Thompson sur les groupes d'ordre impair, C. R. Acad. Sci. Paris Ser. I 229 (1984), 531-534. MR 770439

Similar Articles:

Retrieve articles in Bulletin of the American Mathematical Society with MSC (1985): 20D05, 20-02

Retrieve articles in all Journals with MSC (1985): 20D05, 20-02


Additional Information:

DOI: 10.1090/S0273-0979-1986-15392-9
PII: S 0273-0979(1986)15392-9




AMS and Social Media LinkedIn Facebook Podcasts Twitter YouTube RSS Feeds Blogs Wikipedia