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Finite simple unisingular groups of Lie type. (English) Zbl 1046.20013

Authors’ abstract: Let \(G\) be a finite simple group of Lie type in characteristic \(p\). The authors investigate how large the set of \(p'\)-elements acting without fixed points on an irreducible \(G\)-module in characteristic \(p\) can be. This comes up naturally in studying derangements for finite primitive groups and also in a computational group algorithm proposed by Babai and Shalev. They also consider the same problem for algebraic groups.

MSC:

20C33 Representations of finite groups of Lie type
20G05 Representation theory for linear algebraic groups
20D06 Simple groups: alternating groups and groups of Lie type
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[1] M. Aschbacher. The 27-dimensional module for E6. I. Invent. Math. 89 (1987), 159-195. · Zbl 0629.20018
[2] Aschbacher M., . Math. Soc. 321 pp 45– (1990)
[3] L. Babai and A. Shalev. Recognizing simplicity of black-box groups and the frequency of p-singular elements in a ne groups. In Groups and computation. III (Columbus, OH, 1999), Ohio State Univ. Math. Res. Inst. Publ. 8 (de Gruyter, 2001), pp. 39-62. · Zbl 1052.20031
[4] Borel A., Lecture Notes in Math. pp 131– (1970)
[5] Cameron P. J., Discrete Math. 106 pp 135– (1992)
[6] R. Carter.Finite groups of Lie type: conjugacy classes and complex characters (Wiley, 1985). · Zbl 0567.20023
[7] Cohen A. M., Geom. Dedicata 25 pp 467– (1988)
[8] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson.Atlas of finite groups (Clarendon Press, 1985). · Zbl 0568.20001
[9] Deriziotis D. I., Trans. Amer. Math. Soc. 303 pp 39– (1987)
[10] Dixon J. D., Discrete Math. 105 pp 25– (1992)
[11] Enomoto H., J. Fac. Sci. Univ. Tokyo Sect. I 16 pp 497– (1969)
[12] Fleischmann P., Proc. Amer. Math. Soc. 126 pp 1337– (1998)
[13] Guralnick R. M., J. Algebra 219 pp 345– (1999)
[14] R. M. Guralnick and F. L beck. On p-singular elements in Chevalley groups in characteristic p. In Groups and computation. III (Columbus, OH, 1999), Ohio State Univ. Math. Res. Inst. Publ. 8 (de Gruyter, 2001). pp. 169-182.
[15] Guralnick R. M., Israel J. Math. 101 pp 255– (1997)
[16] Guralnick R. M., Trans. Amer. Math. Soc. 331 pp 563– (1992)
[17] J. E. Humphreys. Introduction toLie algebras and representation theory (Springer-Verlag, 1972). · Zbl 0254.17004
[18] C. Jansen, K. Lux, R. A. Parker and R. A. Wilson.An atlas of Brauer characters (Oxford University Press, 1995). · Zbl 0831.20001
[19] Kleidman P. B., J. Algebra 115 pp 182– (1988)
[20] P. B. Kleidman and M. W. Liebeck.The subgroup structure of the finite classical groups. London Math. Soc. Lecture Note Ser. 129 (Cambridge University Press, 1990). · Zbl 0697.20004
[21] M. W. Liebeck.The a ne permutation groups of rank three. Proc. London Math. Soc. (3) 54 (1987), 477-516. · Zbl 0621.20001
[22] Lang S., Amer. J. Math. 76 pp 819– (1954)
[23] Luczak T., Combin. Probab. Comput. 2 pp 505– (1993)
[24] P. M. Neumann and C. E. Praeger.Derangements and eigenvalue-free elements in finite classical groups. J. London Math. Soc. (2) 58 (1998), 564-586. · Zbl 0936.15020
[25] A. L. Onishchik and E. B. Vinberg.Lie groups and Lie algebras III (Springer-Verlag, 1994). · Zbl 0797.22001
[26] Premet A. A., Math. USSS-Sb. 61 pp 167– (1988)
[27] Springer T. A., Invent. Math. 5 pp 85– (1968)
[28] T. A. Springer.Linear algebraic groups (Birkh user, 1998). · Zbl 0927.20024
[29] Veldkamp F. D., J. Algebra 16 pp 326– (1970)
[30] A. E. Zalesskii. The eigenvalue 1of matrices of complex representations of finite Chevalley groups. Proc. Steklov Inst. Math. (1991), no. 4, 109-119.
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