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.

**[As]**M. Aschbacher, On the maximal subgroups of the finite classical groups,*Invent. Math.***76**(1984), 469-514. MR**86a:20054****[AS]**M. Aschbacher and G.M. Seitz, Involutions in Chevalley groups over finite fields of even order,*Nagoya Math. J.***63**(1976), 1-91. MR**54:10391****[Ba]**L. Babai, On the order of uniprimitive permutation groups,*Annals of Math.***113**(1981), 553-568. MR**83j:20010****[Ca1]**P.J. Cameron, Some open problems on permutation groups, in*Groups, Combinatorics and Geometry*(eds: M.W. Liebeck and J. Saxl), London Math. Soc. Lecture Note Series**165**, Cambridge University Press, Cambridge, 1992, 340-350. MR**94c:20005****[Ca2]**P.J. Cameron, Permutation groups, in*Handbook of Combinatorics*(eds: R.L. Graham et al.), Elsevier Science B.V., Amsterdam, 1995, 611-645. MR**97e:20002****[CK]**P.J. Cameron and W.M. Kantor, Random permutations: some group-theoretic aspects,*Combinatorics, Probability and Computing***2**(1993), 257-262. MR**95b:20006****[CIK]**C.W. Curtis, N. Iwahori and R. Kilmoyer, Hecke algebras and characters of parabolic type of finite groups with -pairs,*IHES Publ. Math.***40**(1972), 81-116. MR**50:494****[Di]**J.D. Dixon, The probability of generating the symmetric group,*Math. Z.***110**(1969), 199-205. MR**40:4985****[GSSh]**D. Gluck, Á. Seress and A. Shalev, Bases for primitive permutation groups and a conjecture of Babai,*J. Algebra***199**(1998), 367-378. CMP**98:06****[GL]**D. Gorenstein and R. Lyons, The local structure of finite groups of characteristic 2 type,*Memoirs Amer. Math. Soc.***42**, No. 276 (1983). MR**84g:20025****[Gu]**R.M. Guralnick, The genus of a permutation group, in*Groups, Combinatorics and Geometry*(eds: M.W. Liebeck and J. Saxl), London Math. Soc. Lecture Note Series**165**(1992), 351-363. MR**94a:20006****[GK]**R.M. Guralnick and W.M. Kantor, Probabilistic generation of finite simple groups,*J. Algebra*, to appear.**[GKS]**R.M. Guralnick, W.M. Kantor and J. Saxl, The probability of generating a classical group,*Comm. in Algebra***22**(1994), 1395-1402. MR**95a:20030****[GLSS]**R.M. Guralnick, M.W. Liebeck, J. Saxl and A. Shalev, Random generation of finite simple groups, to appear.**[GM]**R.M. Guralnick and K. Magaard, On the minimal degree of a primitive permutation group,*J. Algebra***207**(1998), 127-145. CMP**98:17****[GT]**R.M. Guralnick and J.G. Thompson, Finite groups of genus zero,*J. Alg.***131**(1990), 303-341. MR**91e:20006****[HLS]**J. Hall, M.W. Liebeck and G.M. Seitz, Generators for finite simple groups, with applications to linear groups,*Quart. J. Math.***43**(1992), 441-458. MR**93k:20030****[KL]**W.M. Kantor and A. Lubotzky, The probability of generating a finite classical group,*Geom. Ded.***36**(1990), 67-87. MR**91j:20041****[KLi]**P.B. Kleidman and M.W. Liebeck,*The Subgroup Structure of the Finite Classical Groups*, London Math. Soc. Lecture Note Series**129**, Cambridge University Press, 1990. MR**91g:20001****[Li1]**M.W. Liebeck, On the orders of maximal subgroups of the finite classical groups,*Proc. London Math. Soc.***50**(1985), 426-446. MR**87a:20046****[Li2]**M.W. Liebeck, On minimal degrees and base sizes of primitive permutation groups,*Arch. Math.***43**(1984), 11-15. MR**86d:20004****[LP]**M.W. Liebeck and C.W. Purvis, On the genus of a finite classical group,*Bull. London Math. Soc.***29**(1997), 159-164. MR**98h:20086****[LPy]**M.W. Liebeck and L. Pyber, Upper bounds for the number of conjugacy classes of a finite group,*J. Algebra***198**(1997), 538-562. CMP**98:06****[LS]**M.W. Liebeck and J. 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), 266-314. MR**92f:20003****[LiSh1]**M.W. Liebeck and A. Shalev, The probability of generating a finite simple group,*Geom. Ded.***56**(1995), 103-113. MR**96h:20116****[LiSh2]**M.W. Liebeck and A. Shalev, Classical groups, probabilistic methods, and the (2,3)-generation problem,*Annals of Math.***144**(1996), 77-125. MR**97e:20106a****[LPS]**M.W. Liebeck, C.E. Praeger and J. Saxl, On the 2-closures of primitive permutation groups,*J. London Math. Soc.***37**(1988), 241-252. MR**89b:20009****[Ma]**W. Magnus,*Non-Euclidean Tesselations and Their Groups*, Academic Press, New York - London, 1974. MR**50:4774****[Sh1]**A. Shalev, A theorem on random matrices and some applications,*J. Algebra***199**(1998), 124-141. CMP**98:06****[Sh2]**A. Shalev, Random generation of simple groups by two conjugate elements,*Bull. London Math. Soc.***29**(1997), 571-576. MR**98h:20122****[Shi]**T. Shih,*Bounds of Fixed Point Ratios of Permutation Representations of and Groups of Genus Zero*, Ph.D. Thesis, California Institute of Technology, Pasadena, 1990.**[Wa]**G. E. Wall, On the conjugacy classes in the unitary, symplectic and orthogonal groups,*J. Austral. Math. Soc.*3 (1965), 1-62. MR**27:212**

Retrieve articles in *Journal of the American Mathematical Society*
with MSC (1991):
20D06,
20P05

Retrieve articles in all journals with MSC (1991): 20D06, 20P05

Additional Information

**Martin W. Liebeck**

Affiliation:
Department of Mathematics, Imperial College, London SW7 2BZ, England

Email:
m.liebeck@ic.ac.uk

**Aner Shalev**

Affiliation:
Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel

Email:
shalev@math.huji.il

DOI:
https://doi.org/10.1090/S0894-0347-99-00288-X

Received by editor(s):
May 14, 1998

Received by editor(s) in revised form:
August 26, 1998

Additional Notes:
The second author acknowledges the support of the Israel Science Foundation, administered by the Israeli Academy of Sciences and Humanities.

Article copyright:
© Copyright 1999
American Mathematical Society