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

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), no. 3, 469–514. MR**746539**, 10.1007/BF01388470**[AS]**Michael Aschbacher and Gary M. Seitz,*Involutions in Chevalley groups over fields of even order*, Nagoya Math. J.**63**(1976), 1–91. MR**0422401****[Ba]**László Babai,*On the order of uniprimitive permutation groups*, Ann. of Math. (2)**113**(1981), no. 3, 553–568. MR**621016**, 10.2307/2006997**[Ca1]**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**, 10.1017/CBO9780511629259.030**[Ca2]**Peter J. Cameron,*Permutation groups*, Handbook of combinatorics, Vol. 1, 2, Elsevier, Amsterdam, 1995, pp. 611–645. MR**1373668****[CK]**Peter J. Cameron and William M. Kantor,*Random permutations: some group-theoretic aspects*, Combin. Probab. Comput.**2**(1993), no. 3, 257–262. MR**1264032**, 10.1017/S0963548300000651**[CIK]**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**0347996****[Di]**John D. Dixon,*The probability of generating the symmetric group*, Math. Z.**110**(1969), 199–205. MR**0251758****[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]**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**, 10.1090/memo/0276**[Gu]**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**, 10.1017/CBO9780511629259.031**[GK]**R.M. Guralnick and W.M. Kantor, Probabilistic generation of finite simple groups,*J. Algebra*, to appear.**[GKS]**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**, 10.1080/00927879408824912**[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]**Robert M. Guralnick and John G. Thompson,*Finite groups of genus zero*, J. Algebra**131**(1990), no. 1, 303–341. MR**1055011**, 10.1016/0021-8693(90)90178-Q**[HLS]**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**, 10.1093/qmathj/43.4.441**[KL]**William M. Kantor and Alexander Lubotzky,*The probability of generating a finite classical group*, Geom. Dedicata**36**(1990), no. 1, 67–87. MR**1065213**, 10.1007/BF00181465**[KLi]**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****[Li1]**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**, 10.1112/plms/s3-50.3.426**[Li2]**Martin W. Liebeck,*On minimal degrees and base sizes of primitive permutation groups*, Arch. Math. (Basel)**43**(1984), no. 1, 11–15. MR**758332**, 10.1007/BF01193603**[LP]**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**, 10.1112/S0024609396002135**[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]**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**, 10.1112/plms/s3-63.2.266**[LiSh1]**Martin W. Liebeck and Aner Shalev,*The probability of generating a finite simple group*, Geom. Dedicata**56**(1995), no. 1, 103–113. MR**1338320**, 10.1007/BF01263616**[LiSh2]**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**, 10.2307/2118584**[LPS]**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**, 10.1112/jlms/s2-37.2.241**[Ma]**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****[Sh1]**A. Shalev, A theorem on random matrices and some applications,*J. Algebra***199**(1998), 124-141. CMP**98:06****[Sh2]**Aner Shalev,*Random generation of simple groups by two conjugate elements*, Bull. London Math. Soc.**29**(1997), no. 5, 571–576. MR**1458717**, 10.1112/S002460939700338X**[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**(1963), 1–62. MR**0150210**

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:
http://dx.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