A brief history of the classification of the finite simple groups

Solomon, Ronald

doi:10.1090/S0273-0979-01-00909-0

A brief history of the classification of the finite simple groups
HTML articles powered by AMS MathViewer

by Ronald Solomon PDF

Bull. Amer. Math. Soc. 38 (2001), 315-352 Request permission

Abstract:

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

References

J. L. Alperin, Sylow intersections and fusion, J. Algebra 6 (1967), 222–241. MR 215913, DOI 10.1016/0021-8693(67)90005-1
J. L. Alperin, Richard Brauer, and Daniel Gorenstein, Finite groups with quasi-dihedral and wreathed Sylow $2$-subgroups, Trans. Amer. Math. Soc. 151 (1970), 1–261. MR 284499, DOI 10.1090/S0002-9947-1970-0284499-5
J. L. Alperin, Richard Brauer, and Daniel Gorenstein, Finite simple groups of $2$-rank two, Scripta Math. 29 (1973), no. 3-4, 191–214. MR 401902
Michael Aschbacher, Finite groups with a proper $2$-generated core, Trans. Amer. Math. Soc. 197 (1974), 87–112. MR 364427, DOI 10.1090/S0002-9947-1974-0364427-8
Michael Aschbacher, On finite groups of component type, Illinois J. Math. 19 (1975), 87–115. MR 376843
Michael Aschbacher, Tightly embedded subgroups of finite groups, J. Algebra 42 (1976), no. 1, 85–101. MR 422400, DOI 10.1016/0021-8693(76)90028-4
Leo F. Epstein, A function related to the series for $e^{e^x}$, J. Math. Phys. Mass. Inst. Tech. 18 (1939), 153–173. MR 58, DOI 10.1002/sapm1939181153
Michael Aschbacher, Thin finite simple groups, J. Algebra 54 (1978), no. 1, 50–152. MR 511458, DOI 10.1016/0021-8693(78)90022-4
Michael Aschbacher, On the failure of the Thompson factorization in $2$-constrained groups, Proc. London Math. Soc. (3) 43 (1981), no. 3, 425–449. MR 635564, DOI 10.1112/plms/s3-43.3.425
Franz Rádl, Über die Teilbarkeitsbedingungen bei den gewöhnlichen Differential polynomen, Math. Z. 45 (1939), 429–446 (German). MR 82, DOI 10.1007/BF01580293
Rudolph E. Langer, The boundary problem of an ordinary linear differential system in the complex domain, Trans. Amer. Math. Soc. 46 (1939), 151–190 and Correction, 467 (1939). MR 84, DOI 10.1090/S0002-9947-1939-0000084-7

A classification of quasithin groups

Bernd Baumann, Über endliche Gruppen mit einer zu $L_{2}(2^{n})$ isomorphen Faktorgruppe, Proc. Amer. Math. Soc. 74 (1979), no. 2, 215–222 (German). MR 524288, DOI 10.1090/S0002-9939-1979-0524288-0
Helmut Bender, On the uniqueness theorem, Illinois J. Math. 14 (1970), 376–384. MR 262351
Helmut Bender, On groups with abelian Sylow $2$-subgroups, Math. Z. 117 (1970), 164–176. MR 288180, DOI 10.1007/BF01109839
Helmut Bender, Transitive Gruppen gerader Ordnung, in denen jede Involution genau einen Punkt festläßt, J. Algebra 17 (1971), 527–554 (German). MR 288172, DOI 10.1016/0021-8693(71)90008-1
Helmut Bender and George Glauberman, Characters of finite groups with dihedral Sylow $2$-subgroups, J. Algebra 70 (1981), no. 1, 200–215. MR 618388, DOI 10.1016/0021-8693(81)90253-2

Entwicklungslinien in der Theorie endlicher Gruppen

Helmut Bender and George Glauberman, Characters of finite groups with dihedral Sylow $2$-subgroups, J. Algebra 70 (1981), no. 1, 200–215. MR 618388, DOI 10.1016/0021-8693(81)90253-2
Helmut Bender and George Glauberman, Local analysis for the odd order theorem, London Mathematical Society Lecture Note Series, vol. 188, Cambridge University Press, Cambridge, 1994. With the assistance of Walter Carlip. MR 1311244
J. J. Corliss, Upper limits to the real roots of a real algebraic equation, Amer. Math. Monthly 46 (1939), 334–338. MR 4, DOI 10.1080/00029890.1939.11998880
Richard Brauer, Blocks of characters and structure of finite groups, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 1, 21–38. MR 513748, DOI 10.1090/S0273-0979-1979-14548-8
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
Richard Brauer and Michio 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, DOI 10.1073/pnas.45.12.1757
R. Brauer, Michio Suzuki, and G. E. Wall, A characterization of the one-dimensional unimodular projective groups over finite fields, Illinois J. Math. 2 (1958), 718–745. MR 104734, DOI 10.1215/ijm/1255448336

Notes on the theory of groups of finite order

26

On a class of groups of finite order

18

On some properties of groups of odd order

32

On groups of order $p^{\alpha }q^{\beta }$

1

Theory of Groups of Finite Order, Second edition

Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
Michael J. Collins (ed.), Finite simple groups. II, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1980. MR 606048

Simple groups from order $201$ to order $500$

14

Simple groups as far as order $660$

15

On groups whose orders are products of three prime factors

15

E. C. Dade, Lifting group characters, Ann. of Math. (2) 79 (1964), 590–596. MR 160813, DOI 10.2307/1970409
A. Delgado, D. Goldschmidt, and B. Stellmacher, Groups and graphs: new results and methods, DMV Seminar, vol. 6, Birkhäuser Verlag, Basel, 1985. With a preface by the authors and Bernd Fischer. MR 862622

Linear Groups with an Exposition of the Galois Field Theory

Theory of linear groups in an arbitrary field

2

A new system of simple groups

60

Peter Duren and Uta C. Merzbach (eds.), A century of mathematics in America. Part I, History of Mathematics, vol. 1, American Mathematical Society, Providence, RI, 1988. MR 1003156
Michel Enguehard, Obstructions et $p$-groupes de classe $3$, Astérisque 142-143 (1986), 3–48, 296 (French, with English summary). Révision dans les groupes finis. Groupes du type de Lie de rang $1$. MR 873957
Walter Feit, Richard D. Brauer, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 1, 1–20. MR 513747, DOI 10.1090/S0273-0979-1979-14547-6
Walter Feit, Marshall Hall Jr., and John G. Thompson, Finite groups in which the centralizer of any non-identity element is nilpotent, Math. Z. 74 (1960), 1–17. MR 114856, DOI 10.1007/BF01180468
Walter Feit and John G. Thompson, Solvability of groups of odd order, Pacific J. Math. 13 (1963), 775–1029. MR 166261
Bernd Fischer, Finite groups generated by $3$-transpositions. I, Invent. Math. 13 (1971), 232–246. MR 294487, DOI 10.1007/BF01404633

Beiträge zur Theorie der Gruppen endlicher Ordnung

48

Paul Fong and Gary M. Seitz, Groups with a $(B,\,N)$-pair of rank $2$. I, II, Invent. Math. 21 (1973), 1–57; ibid. 24 (1974), 191–239. MR 354858, DOI 10.1007/BF01389689

Über auflösbare Gruppen

Über auflösbare Gruppen IV

Robert H. Gilman and Robert L. Griess Jr., Finite groups with standard components of Lie type over fields of characteristic two, J. Algebra 80 (1983), no. 2, 383–516. MR 691810, DOI 10.1016/0021-8693(83)90007-8
George Glauberman, Central elements in core-free groups, J. Algebra 4 (1966), 403–420. MR 202822, DOI 10.1016/0021-8693(66)90030-5
George Glauberman, A characteristic subgroup of a $p$-stable group, Canadian J. Math. 20 (1968), 1101–1135. MR 230807, DOI 10.4153/CJM-1968-107-2
L. C. Young, On an inequality of Marcel Riesz, Ann. of Math. (2) 40 (1939), 567–574. MR 39, DOI 10.2307/1968941
George Glauberman, On solvable signalizer functors in finite groups, Proc. London Math. Soc. (3) 33 (1976), no. 1, 1–27. MR 417284, DOI 10.1112/plms/s3-33.1.1
George Glauberman and Richard Niles, A pair of characteristic subgroups for pushing-up in finite groups, Proc. London Math. Soc. (3) 46 (1983), no. 3, 411–453. MR 699096, DOI 10.1112/plms/s3-46.3.411
David M. Goldschmidt, A group theoretic proof of the $p^{a}q^{b}$ theorem for odd primes, Math. Z. 113 (1970), 373–375. MR 276338, DOI 10.1007/BF01110506
David M. Goldschmidt, Solvable signalizer functors on finite groups, J. Algebra 21 (1972), 137–148. MR 297861, DOI 10.1016/0021-8693(72)90040-3
David M. Goldschmidt, $2$-signalizer functors on finite groups, J. Algebra 21 (1972), 321–340. MR 323904, DOI 10.1016/0021-8693(72)90027-0
David M. Goldschmidt, $2$-fusion in finite groups, Ann. of Math. (2) 99 (1974), 70–117. MR 335627, DOI 10.2307/1971014
David M. Goldschmidt, Automorphisms of trivalent graphs, Ann. of Math. (2) 111 (1980), no. 2, 377–406. MR 569075, DOI 10.2307/1971203
Daniel Gorenstein, Finite groups, Harper & Row, Publishers, New York-London, 1968. MR 0231903
Daniel Gorenstein, On the centralizers of involutions in finite groups, J. Algebra 11 (1969), 243–277. MR 240188, DOI 10.1016/0021-8693(69)90056-8
Daniel Gorenstein, The classification of finite simple groups. I. Simple groups and local analysis, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 1, 43–199. MR 513750, DOI 10.1090/S0273-0979-1979-14551-8
Daniel Gorenstein, Finite simple groups, University Series in Mathematics, Plenum Publishing Corp., New York, 1982. An introduction to their classification. MR 698782, DOI 10.1007/978-1-4684-8497-7
Daniel Gorenstein, The classification of finite simple groups. Vol. 1, The University Series in Mathematics, Plenum Press, New York, 1983. Groups of noncharacteristic $2$ type. MR 746470, DOI 10.1007/978-1-4613-3685-3
Daniel Gorenstein, The classification of the finite simple groups, a personal journey: the early years, A century of mathematics in America, Part I, Hist. Math., vol. 1, Amer. Math. Soc., Providence, RI, 1988, pp. 447–476. MR 1003189
Daniel Gorenstein and Koichiro Harada, Finite groups whose $2$-subgroups are generated by at most $4$ elements, Memoirs of the American Mathematical Society, No. 147, American Mathematical Society, Providence, R.I., 1974. MR 0367048
Daniel Gorenstein and Richard Lyons, The local structure of finite groups of characteristic $2$ type, Mem. Amer. Math. Soc. 42 (1983), no. 276, vii+731. MR 690900, DOI 10.1090/memo/0276

A local characterization of some sporadic groups

Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, Mathematical Surveys and Monographs, vol. 40, American Mathematical Society, Providence, RI, 1994. MR 1303592, DOI 10.1090/surv/040.1
Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups. Number 3. Part I. Chapter A, Mathematical Surveys and Monographs, vol. 40, American Mathematical Society, Providence, RI, 1998. Almost simple $K$-groups. MR 1490581, DOI 10.1090/surv/040.3
Daniel Gorenstein and John H. Walter, The characterization of finite groups with dihedral Sylow $2$-subgroups. I, J. Algebra 2 (1965), 85–151. MR 177032, DOI 10.1016/0021-8693(65)90027-X
Daniel Gorenstein and John H. Walter, The $\pi$-layer of a finite group, Illinois J. Math. 15 (1971), 555–564. MR 289624
Daniel Gorenstein and John H. Walter, Balance and generation in finite groups, J. Algebra 33 (1975), 224–287. MR 357583, DOI 10.1016/0021-8693(75)90123-4
Robert L. Griess Jr., The friendly giant, Invent. Math. 69 (1982), no. 1, 1–102. MR 671653, DOI 10.1007/BF01389186
Robert L. Griess Jr., Ulrich Meierfrankenfeld, and Yoav Segev, A uniqueness proof for the Monster, Ann. of Math. (2) 130 (1989), no. 3, 567–602. MR 1025167, DOI 10.2307/1971455
K. W. Gruenberg and J. E. Roseblade (eds.), Group theory, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1984. Essays for Philip Hall. MR 780565
Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
Marshall Hall Jr., The theory of groups, The Macmillan Company, New York, N.Y., 1959. MR 0103215

A note on soluble groups

3

A contribution to the theory of groups of prime power order

36

A characteristic property of soluble groups

12

On the Sylow systems of a soluble group

43

Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
Koichiro Harada, Groups with nonconnected Sylow $2$-subgroups revisited, J. Algebra 70 (1981), no. 2, 339–349. MR 623812, DOI 10.1016/0021-8693(81)90222-2

Michio Suzuki

Advanced Studies in Pure Mathematics: Groups and Combinatorics

Die einfachen Gruppen in ersten und zweiten Hundert der Ordnungszahlen

40

Zvonimir Janko, A new finite simple group with abelian Sylow $2$-subgroups and its characterization, J. Algebra 3 (1966), 147–186. MR 193138, DOI 10.1016/0021-8693(66)90010-X

Traité des Substitutions et des Équations Algébriques

Recherches sur les substitutions

17

Kenneth Klinger and Geoffrey Mason, Centralizers of $p$-groups in groups of characteristic $2$, $p$-type, J. Algebra 37 (1975), no. 2, 362–375. MR 390047, DOI 10.1016/0021-8693(75)90084-8
Michael J. Collins (ed.), Finite simple groups. II, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1980. MR 606048

Quasithin finite groups

Hiroshi Matsuyama, Solvability of groups of order $2^{a}p^{b}$, Osaka Math. J. 10 (1973), 375–378. MR 323890
Patrick Paschal McBride, Near solvable signalizer functors on finite groups, J. Algebra 78 (1982), no. 1, 181–214. MR 677717, DOI 10.1016/0021-8693(82)90107-7
Patrick Paschal McBride, Near solvable signalizer functors on finite groups, J. Algebra 78 (1982), no. 1, 181–214. MR 677717, DOI 10.1016/0021-8693(82)90107-7

Theory and Applications of Finite Groups

Proof that there is no simple group whose order lies between $1092$ and $2001$

22

Thomas Peterfalvi, Sur la caractérisation des groupes $\textrm {U}_3(q),$ pour $q$ impair, J. Algebra 141 (1991), no. 2, 253–264 (French). MR 1125694, DOI 10.1016/0021-8693(91)90230-6
Thomas Peterfalvi, Character theory for the odd order theorem, London Mathematical Society Lecture Note Series, vol. 272, Cambridge University Press, Cambridge, 2000. Translated from the 1986 French original by Robert Sandling and revised by the author. MR 1747393, DOI 10.1017/CBO9780511565861
M. B. Powell and G. N. Thwaites, On the non-existence of certain types of subgroups in simple groups, Quart. J. Math. Oxford Ser. (2) 26 (1975), no. 102, 243–256. MR 442078, DOI 10.1093/qmath/26.1.243
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
Rimhak Ree, A family of simple groups associated with the simple Lie algebra of type $(F_{4})$, Amer. J. Math. 83 (1961), 401–420. MR 132781, DOI 10.2307/2372886
Rimhak Ree, A family of simple groups associated with the simple Lie algebra of type $(G_{2})$, Amer. J. Math. 83 (1961), 432–462. MR 138680, DOI 10.2307/2372888
J. E. Roseblade, Obituary: Philip Hall, Bull. London Math. Soc. 16 (1984), no. 6, 603–626. With contributions by J. G. Thompson and J. A. Green. MR 758133, DOI 10.1112/blms/16.6.603
Michael J. Collins (ed.), Finite simple groups. II, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1980. MR 606048
Michael J. Collins (ed.), Finite simple groups. II, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1980. MR 606048
Andreas Speiser, Die Theorie der Gruppen von endlicher Ordnung, 5th ed., Lehrbücher und Monographien aus dem Gebiete der Exakten Wissenschaften (LMW). Mathematische Reihe [Textbooks and Monographs in the Exact Sciences. Mathematical Series], vol. 22, Birkhäuser Verlag, Basel-Boston, Mass., 1980 (German). Mit Anwendungen auf algebraische Zahlen und Gleichungen sowie auf die Kristallographie. [With applications to algebraic numbers and equations and also to crystallography]. MR 607677, DOI 10.1007/978-3-0348-5386-6
Robert Steinberg, Variations on a theme of Chevalley, Pacific J. Math. 9 (1959), 875–891. MR 109191, DOI 10.2140/pjm.1959.9.875
Bernd Stellmacher, A characteristic subgroup of $\Sigma _4$-free groups, Israel J. Math. 94 (1996), 367–379. MR 1394582, DOI 10.1007/BF02762712
Bernd Stellmacher, An application of the amalgam method: the $2$-local structure of $N$-groups of characteristic $2$ type, J. Algebra 190 (1997), no. 1, 11–67. MR 1442145, DOI 10.1006/jabr.1996.6864

Finite groups all of whose maximal parabolic subgroups are thin and of characteristic $2$-type

C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
Michio Suzuki, A new type of simple groups of finite order, Proc. Nat. Acad. Sci. U.S.A. 46 (1960), 868–870. MR 120283, DOI 10.1073/pnas.46.6.868
Michio Suzuki, On a class of doubly transitive groups, Ann. of Math. (2) 75 (1962), 105–145. MR 136646, DOI 10.2307/1970423
Michio Suzuki, On a class of doubly transitive groups. II, Ann. of Math. (2) 79 (1964), 514–589. MR 162840, DOI 10.2307/1970408

Théorèmes sur les groupes de substitutions

5

Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
John G. Thompson, Normal $p$-complements for finite groups, Math. Z 72 (1959/1960), 332–354. MR 0117289, DOI 10.1007/BF01162958
Frederic H. Miller, Calculus, John Wiley & Sons, Inc., New York, 1939. MR 0000037
J. G. Thompson, Quadratic pairs, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 375–376. MR 0430043
K. W. Gruenberg and J. E. Roseblade (eds.), Group theory, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1984. Essays for Philip Hall. MR 780565
F. G. Timmesfeld, Groups generated by root-involutions. I, II, J. Algebra 33 (1975), 75–134; ibid. 35 (1975), 367–441. MR 372019, DOI 10.1016/0021-8693(75)90133-7
Franz Timmesfeld, Finite simple groups in which the generalized Fitting group of the centralizer of some involution is extraspecial, Ann. of Math. (2) 107 (1978), no. 2, 297–369. MR 486255, DOI 10.2307/1971146
Jacques Tits, Buildings of spherical type and finite BN-pairs, Lecture Notes in Mathematics, Vol. 386, Springer-Verlag, Berlin-New York, 1974. MR 0470099
John H. Walter, The characterization of finite groups with abelian Sylow $2$-subgroups, Ann. of Math. (2) 89 (1969), 405–514. MR 249504, DOI 10.2307/1970648
Leonard Eugene Dickson, New First Course in the Theory of Equations, John Wiley & Sons, Inc., New York, 1939. MR 0000002
Tomoyuki Yoshida, Character-theoretic transfer, J. Algebra 52 (1978), no. 1, 1–38. MR 491920, DOI 10.1016/0021-8693(78)90259-4

Kennzeichnung endlicher linearer Gruppen

11

Über endliche Fastkörper

11

Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944