Simple groups, permutation groups, and probability

Liebeck, Martin; Shalev, Aner

doi:10.1090/S0894-0347-99-00288-X

Simple groups, permutation groups, and probability

Authors: Martin W. Liebeck and Aner Shalev
Journal: J. Amer. Math. Soc. 12 (1999), 497-520
MSC (1991): Primary 20D06; Secondary 20P05
DOI: https://doi.org/10.1090/S0894-0347-99-00288-X
MathSciNet review: 1639620
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We derive a new bound for the minimal degree of an almost simple primitive permutation group, and settle a conjecture of Cameron and Kantor concerning the base size of such a group. Additional results concern random generation of simple groups, and the so-called genus conjecture of Guralnick and Thompson. Our proofs are based on probabilistic arguments, together with a new result concerning the size of the intersection of a maximal subgroup of a classical group with a conjugacy class of elements.

M. Aschbacher, On the maximal subgroups of the finite classical groups, Invent. Math. 76 (1984), no. 3, 469–514. MR 746539, DOI https://doi.org/10.1007/BF01388470
Michael Aschbacher and Gary M. Seitz, Involutions in Chevalley groups over fields of even order, Nagoya Math. J. 63 (1976), 1–91. MR 422401
László Babai, On the order of uniprimitive permutation groups, Ann. of Math. (2) 113 (1981), no. 3, 553–568. MR 621016, DOI https://doi.org/10.2307/2006997
Peter J. Cameron, Some open problems on permutation groups, Groups, combinatorics & geometry (Durham, 1990) London Math. Soc. Lecture Note Ser., vol. 165, Cambridge Univ. Press, Cambridge, 1992, pp. 340–350. MR 1200272, DOI https://doi.org/10.1017/CBO9780511629259.030
Peter J. Cameron, Permutation groups, Handbook of combinatorics, Vol. 1, 2, Elsevier Sci. B. V., Amsterdam, 1995, pp. 611–645. MR 1373668
Peter J. Cameron and William M. Kantor, Random permutations: some group-theoretic aspects, Combin. Probab. Comput. 2 (1993), no. 3, 257–262. MR 1264032, DOI https://doi.org/10.1017/S0963548300000651
C. W. Curtis, N. Iwahori, and R. Kilmoyer, Hecke algebras and characters of parabolic type of finite groups with $(B,$ $N)$-pairs, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 81–116. MR 347996
John D. Dixon, The probability of generating the symmetric group, Math. Z. 110 (1969), 199–205. MR 251758, DOI https://doi.org/10.1007/BF01110210

D. Gluck, Á. Seress and A. Shalev, Bases for primitive permutation groups and a conjecture of Babai, J. Algebra 199 (1998), 367-378.

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 https://doi.org/10.1090/memo/0276

Robert M. Guralnick, The genus of a permutation group, Groups, combinatorics & geometry (Durham, 1990) London Math. Soc. Lecture Note Ser., vol. 165, Cambridge Univ. Press, Cambridge, 1992, pp. 351–363. MR 1200273, DOI https://doi.org/10.1017/CBO9780511629259.031

R.M. Guralnick and W.M. Kantor, Probabilistic generation of finite simple groups, J. Algebra, to appear.

Robert M. Guralnick, William M. Kantor, and Jan Saxl, The probability of generating a classical group, Comm. Algebra 22 (1994), no. 4, 1395–1402. MR 1261266, DOI https://doi.org/10.1080/00927879408824912

R.M. Guralnick, M.W. Liebeck, J. Saxl and A. Shalev, Random generation of finite simple groups, to appear.

R.M. Guralnick and K. Magaard, On the minimal degree of a primitive permutation group, J. Algebra 207 (1998), 127-145.

Robert M. Guralnick and John G. Thompson, Finite groups of genus zero, J. Algebra 131 (1990), no. 1, 303–341. MR 1055011, DOI https://doi.org/10.1016/0021-8693%2890%2990178-Q

J. I. Hall, Martin W. Liebeck, and Gary M. Seitz, Generators for finite simple groups, with applications to linear groups, Quart. J. Math. Oxford Ser. (2) 43 (1992), no. 172, 441–458. MR 1188385, DOI https://doi.org/10.1093/qmathj/43.4.441

William M. Kantor and Alexander Lubotzky, The probability of generating a finite classical group, Geom. Dedicata 36 (1990), no. 1, 67–87. MR 1065213, DOI https://doi.org/10.1007/BF00181465

Peter Kleidman and Martin Liebeck, The subgroup structure of the finite classical groups, London Mathematical Society Lecture Note Series, vol. 129, Cambridge University Press, Cambridge, 1990. MR 1057341

Martin W. Liebeck, On the orders of maximal subgroups of the finite classical groups, Proc. London Math. Soc. (3) 50 (1985), no. 3, 426–446. MR 779398, DOI https://doi.org/10.1112/plms/s3-50.3.426

Martin W. Liebeck, On minimal degrees and base sizes of primitive permutation groups, Arch. Math. (Basel) 43 (1984), no. 1, 11–15. MR 758332, DOI https://doi.org/10.1007/BF01193603

Martin W. Liebeck and Chris Wayman Purvis, On the genus of a finite classical group, Bull. London Math. Soc. 29 (1997), no. 2, 159–164. MR 1425992, DOI https://doi.org/10.1112/S0024609396002135

M.W. Liebeck and L. Pyber, Upper bounds for the number of conjugacy classes of a finite group, J. Algebra 198 (1997), 538-562.

Martin W. Liebeck and Jan Saxl, Minimal degrees of primitive permutation groups, with an application to monodromy groups of covers of Riemann surfaces, Proc. London Math. Soc. (3) 63 (1991), no. 2, 266–314. MR 1114511, DOI https://doi.org/10.1112/plms/s3-63.2.266

Martin W. Liebeck and Aner Shalev, The probability of generating a finite simple group, Geom. Dedicata 56 (1995), no. 1, 103–113. MR 1338320, DOI https://doi.org/10.1007/BF01263616

Martin W. Liebeck and Aner Shalev, Classical groups, probabilistic methods, and the $(2,3)$-generation problem, Ann. of Math. (2) 144 (1996), no. 1, 77–125. MR 1405944, DOI https://doi.org/10.2307/2118584

Martin W. Liebeck, Cheryl E. Praeger, and Jan Saxl, On the $2$-closures of finite permutation groups, J. London Math. Soc. (2) 37 (1988), no. 2, 241–252. MR 928521, DOI https://doi.org/10.1112/jlms/s2-37.2.241

Wilhelm Magnus, Noneuclidean tesselations and their groups, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Pure and Applied Mathematics, Vol. 61. MR 0352287

A. Shalev, A theorem on random matrices and some applications, J. Algebra 199 (1998), 124-141.

Aner Shalev, Random generation of simple groups by two conjugate elements, Bull. London Math. Soc. 29 (1997), no. 5, 571–576. MR 1458717, DOI https://doi.org/10.1112/S002460939700338X

T. Shih, Bounds of Fixed Point Ratios of Permutation Representations of $GL_n(q)$ and Groups of Genus Zero, Ph.D. Thesis, California Institute of Technology, Pasadena, 1990.

G. E. Wall, On the conjugacy classes in the unitary, symplectic and orthogonal groups, J. Austral. Math. Soc. 3 (1963), 1–62. MR 0150210