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Generators and Relations in Groups and Geometries pp 195–233Cite as

2-Generation of finite simple groups and some related topics

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Part of the book series: NATO ASI Series ((ASIC,volume 333))

Abstract

It is well-known that every finite simple group is 2-generated, i.e. it can be generated by two suitable elements. This is the topic discussed in §1, which centers around Steinberg’s unified treatment of groups of Lie type. In §2 we discuss generation of simple groups by special kinds of generating pairs, namely: 1) the generation of simple groups of Lie type by a cyclic maximal torus and a long root element, with application to the solution of the Magnus-Gorchakov-Levchuk conjecture on residual properties of free groups; 2) the generation of a simple group by an involution and another suitable element. With regard to 1), we also mention similar 2-generation results in connection with Galois groups; with regard to 2), emphasis is put on (2,3)-generation and Hurwitz generation of finite simple groups. Finally, §3 deals with generating sets of involutions of minimal size. Most finite simple groups are generated by three involutions. Generation results, a non-generation criterion, and a relation between (2,3)-generation and generation by three involutions are illustrated.

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References

  1. Albert A.A., Thompson J.G. ”Two-element generation of the projective unimodular group” Illinois Math. J. 3 (1959), 421–439.

    MathSciNet  MATH  Google Scholar 

  2. Aschbacher M. ”Finite Groups Generated by Odd Transpositions I”, Math. Z. 127 (1972), 46–56.

    Article  MathSciNet  Google Scholar 

  3. Aschbacher M. ”Finite Groups Generated by Odd Transpositions II, III, IV”, J. Algebra 26 (1973), 451–459, 460–478, 479–491.

    Article  MathSciNet  MATH  Google Scholar 

  4. Aschbacher M. ”On the maximal subgroups of the finite classical groups” Invent. Math. 76 (1984), 469–514.

    Article  MathSciNet  MATH  Google Scholar 

  5. Aschbacher M., Guralnick R. ”Solvable generation of groups and Sylow subgroups of the lower central series”, J. Algebra 77 (1982), 189–201.

    Article  MathSciNet  MATH  Google Scholar 

  6. Aschbacher M., Guralnick R. ”Some applications of the first cohomology group” J. Algebra 90 (1984), 446–460.

    Article  MathSciNet  MATH  Google Scholar 

  7. Belyî G.V. ”On Galois extensions of a maximal cyclotomic field”, Math. USSR Izvestija, AMS Trans. 14 (1980), No.2, 247–256.

    Article  MATH  Google Scholar 

  8. Bovey J., Williamson A. ”The probability of generating the symmetric group” Bull. London Math. Soc. 10 (1978), 91–96.

    Article  MathSciNet  MATH  Google Scholar 

  9. Brahana H.R. ”Regular maps and their groups” Amer. J. Math. 49 (1927), 268–284.

    Article  MathSciNet  MATH  Google Scholar 

  10. Brahana H.R. ”Certain perfect groups generated by two operators of orders two and three” Amer. J. Math. 50 (1928), 345–356.

    Article  MathSciNet  MATH  Google Scholar 

  11. Brahana H.R. ”Pairs of generators of the known simple groups whose orders are less than one million” Ann. of Math. 31 (1930), 529–549.

    Article  MathSciNet  MATH  Google Scholar 

  12. Brahana H.R. ”On the groups generated by two operators of order two and three whose product is of order 8” Amer. J. Math. 53 (1931), 891–901.

    Google Scholar 

  13. Brenner J.L., Guralnick R.M. Wiegold J. ”Two generator groups III” Amer. Math. Soc. 33 (1984), 82–89.

    MathSciNet  Google Scholar 

  14. Brenner J.L., Wiegold J. ”Two generator groups I” Michigan Math. J. 22 (1975), 53–64.

    Article  MathSciNet  MATH  Google Scholar 

  15. Brenner J.L., Wiegold J. ”Two generator groups II” Bull. Austral. Math. Soc. 22 (1980), 113–124.

    Article  MathSciNet  MATH  Google Scholar 

  16. Burnside W. ”Theory of Groups of Finite Order” 2nd ed., Cambridge University Press (1911) (Dover Reprint 1955).

    Google Scholar 

  17. Bussey W.H. ”Generational relations for the abstract group simply isomorphic with the group LF[2, p n]”, Proc. London Math. Soc. (2), 3 (1905), 296–315.

    Article  MathSciNet  MATH  Google Scholar 

  18. Carmichael R.D. ”Abstract definitions of the symmetric and alternating groups and certain other permutation groups”, Quart. J. Math. 49, 226–270.

    Google Scholar 

  19. Carter R.W. ”Simple groups of Lie type”, J. Wiley and Sons (1972).

    Google Scholar 

  20. Conder M.D.E. ”Generators for alternating and symmetric groups” J. London Math. Soc. (2) 22 (1980), 75–86.

    Article  MathSciNet  MATH  Google Scholar 

  21. Conder M.D.E. ”More on generators for alternating and symmetric groups” Quart. J. Math. Oxford (2) 32 (1981), 137–163.

    Article  MathSciNet  MATH  Google Scholar 

  22. Conder M.D.E. ”The symmetric genus of Alternating and Symmetric groups” J. Comb. Theory B39 (1985), 179–186.

    MathSciNet  Google Scholar 

  23. Coxeter H.S.M. ”An abstract definition for the alternating group in terms of two generators”, J. London Math. Soc. 11. (1936), 150–156.

    Article  MathSciNet  Google Scholar 

  24. Coxeter H.S.M., Moser W.O.J. ”Generators and Relations for Discrete Groups (Fourth Edition)” Springer Verlag, (1980).

    Google Scholar 

  25. Dalla Volta F. ”Gruppi sporadici generati da tre involuzioni” RILS A 119 (1985), 65–87.

    MathSciNet  Google Scholar 

  26. Dalla Volta F., Tamburini M.C. ”Generazione di Psp (4,q) mediante tre involuzioni” Boll. Un. Mat. Ital. (7) 3-A (1989), 285–289.

    Google Scholar 

  27. Dalla Volta F., Tamburini M.C. ”Generation of some orthogonal groups by a set of 3 involutions” (to appear).

    Google Scholar 

  28. Dey I.M.S., Wieglod J. ”Generators for alternating and symmetric groups” J. Austral. Math. Soc. 12 (1971), 63–68.

    Article  MathSciNet  MATH  Google Scholar 

  29. Dyck W. ”Gruppentheoretische Studien” Math. Ann. 20 (1882), 1–45.

    Article  MathSciNet  Google Scholar 

  30. Dickson L.E. ”Linear Groups, with an Exposition of the Galois Field Theory”, Teubner, Leipzig (1901) (Dover Reprint 1958).

    MATH  Google Scholar 

  31. Di Martino L. ”Simple linear groups all of whose involutions are 2-reflections” Boll. Un. Mat. Ital. (5) 15-B (1978), 509–526.

    Google Scholar 

  32. Farkas H.M., Kra I. ”Riemann Surfaces” Springer Verlag New York Heidelberg Berlin (1980).

    Book  MATH  Google Scholar 

  33. Feit W. ”Rigidity and Galois groups”, in Proceedings of the Rutgers Group Theory Year 1983–1984, ed. Aschbacher et al., Cambridge University Press (1984), 283–287.

    Google Scholar 

  34. Frasch H. ”Die Erzeugenden der Hauptkongruenzgruppen für Primzahlstufen”, Math. Ann. 108 (1933), 229–252.

    Article  MathSciNet  Google Scholar 

  35. Fricke R., Klein F. ”Vorlesungen über die Theorie der Elliptischen Modul funktionen” Vol. 1,2; Teubner, Leipzig (1890).

    Google Scholar 

  36. Gillio Berta Mauri A., Tamburini M.C. ”Alcune classi di gruppi generati da tre involuzioni” RILS A116 (1982), 191–209.

    MathSciNet  Google Scholar 

  37. Glover H., Sjerve D. ”Representing PSL 2(P) on a Riemann surface of least genus” L’Enseignement Mathématique 31 (1985), 305–325.

    MathSciNet  MATH  Google Scholar 

  38. Gorenstein D. ”Finite Groups”, Harper and Row (1968).

    Google Scholar 

  39. Gorenstein D. ”Finite Simple Groups. An Introduction to their Classification” Plenum Press, New York and London (1982).

    MATH  Google Scholar 

  40. Gow R. ”Products of two involutions in classical groups of characteristic 2” J. Algebra 71 (1981), 583–591.

    Article  MathSciNet  MATH  Google Scholar 

  41. Guralnick R.M. ”Generation of simple groups” J. Algebra 103 (1986), 381–401.

    Article  MathSciNet  MATH  Google Scholar 

  42. Guralnick R.M. ”On the number of generators of a finite group” Arch. Math. 53 (1989), 521–523.

    Article  MathSciNet  MATH  Google Scholar 

  43. Hestenes M.D. ”Singer Groups”, Can. J. Math., 22(3) (1970), 492–513.

    Article  MathSciNet  MATH  Google Scholar 

  44. Humphreys J.E. ”Introduction to Lie Algebras and Representation Theory” Springer Verlag (1972).

    Google Scholar 

  45. Huppert B. ”Singerzykel in klassischen gruppen” Math. Z. 117 (1970), 141–150.

    Article  MathSciNet  MATH  Google Scholar 

  46. Hurwitz A. ”Über algebraische Gebilde mit eindeutigen Transformationen in sich” Math. Ann. 41 (1893), 408–442.

    Google Scholar 

  47. Ito N. ”On a certain class of connected symmetric trivalent graphs” Math. J. Okayama Univ. 25 (1983), 145–152.

    MathSciNet  MATH  Google Scholar 

  48. Jacobson N. ”Lie Algebras” Wiley Interscience, New York-London (1962).

    MATH  Google Scholar 

  49. Jacobson N. ”Exceptional Lie Algebras” M. Dekker, New York (1971).

    MATH  Google Scholar 

  50. Kantor W. ”Subgroups of classical groups generated by long root elements” Trans. Amer. Math. Soc. 248 (1979), 347–379.

    Article  MathSciNet  MATH  Google Scholar 

  51. Katz R., Magnus W. ”Residual properties of free groups” Comm. Pure Appl. Math. 22 (1969), 1–13.

    Article  MathSciNet  MATH  Google Scholar 

  52. Klein F. ”Uber die Transformationen siebenter Ordnung der elliptischen Funktionen”, Math. Ann. 14 (1879), 428–471.

    Article  MATH  Google Scholar 

  53. Kovàcs L.G. ”On finite solvable groups” Math. Z. 103 (1968), 37–39.

    Article  MathSciNet  MATH  Google Scholar 

  54. Levchuk V.M. ”A Property of Suzuki groups”, Alg. i Log. 11 (1972), 551–557.

    MATH  Google Scholar 

  55. Levchuk V.M., Nuzhin Y.N. ”Structure of Ree groups”, Alg. i Log. 24 (1985), 26–41.

    MathSciNet  Google Scholar 

  56. Longobardi P., Maj M. ”On the number of generators of a finite group” Arch. Math. 50 (1988), 110–112.

    Article  MathSciNet  MATH  Google Scholar 

  57. Lorimer P. ”Embedding of symmetric graphs in surfaces”, to appear.

    Google Scholar 

  58. Lubotzky A. ”On a problem of Magnus” Proc. Amer. Math. Soc. (4) 98 (1986).

    Google Scholar 

  59. Lucchini A. ”A bound on the number of generators of a finite group”, Arch. Math. 53 (1989), 313–317.

    Article  MathSciNet  MATH  Google Scholar 

  60. Lucchini A. ”Some questions on the number of generators of a finite group”, to appear in Rend. Sem. Mat. Padova.

    Google Scholar 

  61. Macbeath A.M. ”Generators of linear fractional groups” Proc. Symp. Pure Math. 12 (1969), 14–32.

    MathSciNet  Google Scholar 

  62. Magnus W. ”Residually finite groups” Bull. Amer. Math. Soc. 75 (1969), 306–316.

    Article  MathSciNet  Google Scholar 

  63. Magnus W. ”Noneuclidean Tesselations and their Groups”, Academic Press (1974).

    Google Scholar 

  64. Miller G.A. ”On the groups generated by two operators” Bull. AMS 7 (1901), 424–426.

    Article  MATH  Google Scholar 

  65. Miller G.A. ”On the groups generated by two operators of orders two and three respectively whose product is of order six” Quarterly J. of Math. 33 (1901), 76–79.

    Google Scholar 

  66. Miller G.A. ”Groups defined by the orders of two generators and the order of their product” Amer. J. Math. 24 (1902), 96–100.

    Article  MathSciNet  MATH  Google Scholar 

  67. Miller G.A. ”Possible orders of two generators of the alternating and of the symmetric group” Trans. AMS 30 (1928), 24–32.

    MATH  Google Scholar 

  68. Newman M. ”Maximal normal subgroups of the modular group” Proc. Amer. Math. Soc. 19 (1968), 1138–1144.

    Article  MathSciNet  MATH  Google Scholar 

  69. Peluso A. ”A residual property of free groups” Comm. Pure Appl. Math. 19 (1966), 435–437.

    Article  MathSciNet  MATH  Google Scholar 

  70. Piper F.C. ”On elations of finite projective spaces of odd order” J. London Math. Soc. 41 (1966), 641–648.

    Article  MathSciNet  MATH  Google Scholar 

  71. Piper F.C. ”On elations of finite projective spaces of even order” J. London Math. Soc. 43 (1968), 456–464.

    Article  MathSciNet  Google Scholar 

  72. Pride S.J. ”Residual properties of free groups I, II, III” Pacific J. Math. 43 (1972), 725–733; Bull. Austral. Math. Soc. 7 (1972), 113–120; Math. Z. 132 (1972), 245–248.

    Article  MathSciNet  MATH  Google Scholar 

  73. Ree R. ”On some simple groups defined by Chevalley” Trans. Amer. Math. Soc. 84 (1957), 392–400.

    Article  MathSciNet  MATH  Google Scholar 

  74. Room T.G. ”The generation by two operators of the symplectic group over GF(2)” J. Austral. Math. Soc. 1 (1959), 38–46.

    Article  MathSciNet  MATH  Google Scholar 

  75. Room T.G., Smith R.J. ”A generation of the symplectic group” Quart. J. Math. Oxford Ser. (2) 9 (1958), 177–182.

    Article  MathSciNet  MATH  Google Scholar 

  76. Salzberg Stark B. ”Irreducible subgroups of the orthogonal groups generated by groups of root type I” Pacific J. Math. 53 (1974), 611.

    Article  MathSciNet  MATH  Google Scholar 

  77. Serezkin V.N. ”Reflection Groups over Finite Fields of Characteristic p > 5”, Soviet Math. Dokl. 17 (1976), 478–480.

    MATH  Google Scholar 

  78. Serezkin V.N., Zalesskiî A.E. ”Finite linear groups generated by reflections” Math. USSR Izvestija, AMS Trans. 17, 477–503.

    Google Scholar 

  79. Sinkov A. ”Necessary and sufficient conditions for generating certain simple groups by two operators of periods two and three” Amer. J. Math. 59 (1937) 67–76.

    Article  MathSciNet  Google Scholar 

  80. Sinkov A. ”On the group-defining relations (2,3,7;p)” Ann. of Math. 38 (1937), 577–584.

    Article  MathSciNet  Google Scholar 

  81. Sinkov A. ”On generating the simple group LF (2,2n), by two operators of periods two and three” Bull. Amer. Math. Soc. 44 (1938), 449–455.

    Article  MathSciNet  Google Scholar 

  82. Sinkov A. ”The number of abstract definitions of LF(2,p) as a quotient group of (2,3,n)”, J. Algebra 12 (1969), 525–532.

    Article  MathSciNet  MATH  Google Scholar 

  83. Stanek P. ”Two-element generation of the symplectic group” Trans. Amer. Math. Soc. 108 (1963), 429–436.

    Article  MathSciNet  MATH  Google Scholar 

  84. Steinberg R. ”Generators for simple groups” Canad. J. Math. 14 (1962), 277–283.

    Article  MathSciNet  MATH  Google Scholar 

  85. Steinberg R. ”Lectures on Chevalley groups” Yale University Mimeographed Notes.

    Google Scholar 

  86. Tamburini M.C. ”Generation of certain simple groups by elements of small order” RILS A 121 (1987), 21–27.

    MathSciNet  Google Scholar 

  87. Tamburini M.C., Wilson J.S. ”A Residual property of certain free products”, Math. Z. 186 (1984), 525–530.

    Article  MathSciNet  MATH  Google Scholar 

  88. Tamburini M.C., Wilson J.S. ”On the generation of finite simple groups by pairs of subgroups” J. Algebra (2) 116 (1988), 316–333.

    Article  MathSciNet  MATH  Google Scholar 

  89. Thompson J.G. ”Quadratic Pairs”, Actes, Conprès Intern. Math., Tome I (1970), 375–376.

    Google Scholar 

  90. Tucker T.W. ”Finite groups acting on surfaces and the genus of a group” J. Comb. Theory B 34 (1983), 82–98.

    Article  MATH  Google Scholar 

  91. Wagner A. ”Groups generated by elations” Abh. Hamburg 41. (1974), 199–205.

    Article  Google Scholar 

  92. Wagner A. ”Collineation groups generated by homologies of order greater than 2”, Geom. Dedicata 7, 387–398.

    Google Scholar 

  93. Wagner A. ”The minimal number of involutions generating some finite three-dimensional groups” Boll. Un. Mat. Ital. (5) 15-A (1978), 431–439.

    Google Scholar 

  94. Wagner A. ”Determination of the finite primitive reflection groups over an arbitrary field of characteristic not 2”, Geom. Dedicata, I, 9 (1980), 239–253; II, 10 (1981), 183–189; III, 10 (1981), 475–523.

    MathSciNet  MATH  Google Scholar 

  95. Walter J.H. ”Classical groups as Galois groups”, in Proceedings of the Rutgers Group Theory Year 1983–1984, ed. Aschbacher et al., Cambridge University Press (1984), 357–383.

    Google Scholar 

  96. Weigel T. ”Residuelle Eigenschaften freier Gruppen” (1987) Diplomarbeit.

    Google Scholar 

  97. Weigel T. ”Residuelle Eigenschaften freier Gruppen” (1989) Dissertation (Freiburg).

    Google Scholar 

  98. Wiegold J. ”Free groups residually alternating of even degree” Arch. Math. (Basel) 28 (1977), 337–339.

    Article  MathSciNet  MATH  Google Scholar 

  99. Wielandt H. ”Einbettung zweier Gruppen in eine einfache Gruppe”, Math. Z. 73 (1960), 20–21.

    Article  MathSciNet  MATH  Google Scholar 

  100. Wielandt H. ”Finite permutation groups” Academic Press (1964).

    Google Scholar 

  101. Wilson J.S. ”A residual property of free groups” (to appear in J. Algebra).

    Google Scholar 

  102. Woldar A.J. ”On Hurwitz generation and genus actions of sporadic groups” Illinois Math. J. (3) 33 (1989), 416–437.

    MathSciNet  MATH  Google Scholar 

  103. Woldar A.J. ”Genus action of finite simple groups” Illinois Math. J. (3) 33 (1989), 438–450.

    MathSciNet  MATH  Google Scholar 

  104. Woldar A.J. ”Representing the Mathien groups on surfaces at least genus”, to appear.

    Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica „F. Enriques“, Università degli Studi di Milano, Via C. Saldini, 50, 20133, Milano, Italy

    L. Di Martino

  2. Dipartimento di Matematica, Università Cattolica del Sacro Cuore, Via Trieste, 17, 25121, Brescia, Italy

    M. C. Tamburini

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  1. L. Di Martino
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  2. M. C. Tamburini
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Editors and Affiliations

  1. Dipartimento di Matematica “Ulisse Dini”, Università degli Studi di Firenze, Florence, Italy

    A. Barlotti

  2. Department of Mathematics, University of Toronto, Toronto, Ontario, Canada

    E. W. Ellers

  3. Mathematisches Institut, Universität Erlangen-Nürnberg, Erlangen, Germany

    P. Plaumann  & K. Strambach  & 

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Di Martino, L., Tamburini, M.C. (1991). 2-Generation of finite simple groups and some related topics. In: Barlotti, A., Ellers, E.W., Plaumann, P., Strambach, K. (eds) Generators and Relations in Groups and Geometries. NATO ASI Series, vol 333. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-3382-1_8

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