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On the uniform spread of almost simple linear groups

Published online by Cambridge University Press:  11 January 2016

Timothy C. Burness  and
Simon Guest
Timothy C. Burness
Affiliation:
School of Mathematics University of Southampton, Southampton SO17 1BJ, UK, [email protected]
Simon Guest
Affiliation:
School of Mathematics University of Southampton, Southampton SO17 1BJ, UK, [email protected]
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Abstract

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Let G be a finite group, and let k be a nonnegative integer. We say that G has uniform spread k if there exists a fixed conjugacy class C in G with the property that for any k nontrivial elements x1,…,xk in G there exists y ∊ C such that G = ‹xi,y› for all i. Further, the exact uniform spread of G, denoted by u(G), is the largest k such that G has the uniform spread k property. By a theorem of Breuer, Guralnick, and Kantor, u(G) ≥ 2 for every finite simple group G. Here we consider the uniform spread of almost simple linear groups. Our main theorem states that if G = ‹PSLn (q),g› is almost simple, then u(G) ≥ 2 (unless G ≅ S6), and we determine precisely when u(G) tends to infinity as |G| tends to infinity.

Keywords

20D0620E2820F0520P05
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 209 , March 2013 , pp. 35 - 109
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2013

References

[1] Aschbacher, M., On the maximal subgroups of the finite classical groups, Invent. Math. 76 (1984), 469–514. MR 0746539. DOI 10.1007/BF01388470.Google Scholar
[2] Aschbacher, M. and Seitz, G. M., Involutions in Chevalley groups over fields of even order, Nagoya Math. J. 63 (1976), 1–91. MR 0422401.CrossRefGoogle Scholar
[3] Binder, G. J., The two-element bases of the symmetric group, Izv. Vysš. Učebn. Zaved. Mat. 90 (1970), 9–11. MR 0257197.Google Scholar
[4] Bosma, W. and Cannon, J. J., Handbook of Magma Functions, School of Mathematics and Statistics, University of Sydney, Sydney, 1995.Google Scholar
[5] Bray, J. N., Holt, D. F., and Roney–Dougal, C. M., The Maximal Subgroups of the Low-Dimensional Finite Classical Groups, to appear in London Math. Soc. Lecture Note Ser., Cambridge University Press.Google Scholar
[6] Brenner, J. L. and Wiegold, J., Two-generator groups, I, Michigan Math. J. 22 (1975), 53–64. MR 0372033.CrossRefGoogle Scholar
[7] Breuer, T., Guralnick, R. M., and Kantor, W. M., Probabilistic generation of finite simple groups, II, J. Algebra 320 (2008), 443–494. MR 2422303. DOI 10.1016/j. jalgebra.2007.10.028.Google Scholar
[8] Breuer, T., Guralnick, R. M., Lucchini, A., Maróti, A., and Nagy, G. P., Hamiltonian cycles in the generating graph of finite groups, Bull. Lond. Math. Soc. 42 (2010), 621–633. MR 2669683. DOI 10.1112/blms/bdq017.CrossRefGoogle Scholar
[9] Burness, T. C., Fixed point ratios in actions of finite classical groups, I, J. Algebra 309 (2007), 69–79. MR 2301233. DOI 10.1016/j.jalgebra.2006.05.024.Google Scholar
[10] Burness, T. C., Fixed point ratios in actions of finite classical groups, II, J. Algebra 309 (2007), 80–138. MR 2301234. DOI 10.1016/j.jalgebra.2006.05.025.Google Scholar
[11] Burness, T. C., Fixed point ratios in actions of finite classical groups, III, J. Algebra 314 (2007), 693–748. MR 2344583. DOI 10.1016/j.jalgebra.2007.01.011.Google Scholar
[12] Burness, T. C., Fixed point ratios in actions of finite classical groups, IV, J. Algebra 314 (2007), 749–788. MR 2344584. DOI 10.1016/j.jalgebra.2007.01.012.Google Scholar
[13] Cannon, J. J. and Holt, D. F., Automorphism group computation and isomorphism testing in finite groups, J. Symbolic Comput. 35 (2003), 241–267. MR 1962794. DOI 10.1016/S0747-7171(02)00133-5.Google Scholar
[14] Cannon, J. J. and Holt, D. F., Computing maximal subgroups of finite groups, J. Symbolic Comput. 37 (2004), 589–609. MR 2094616. DOI 10.1016/j.jsc.2003.08.002.Google Scholar
[15] Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A., and Wilson, R. A., Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups, Oxford University Press, Eynsham, 1985. MR 0827219.Google Scholar
[16] Volta, F. Dalla and Lucchini, A., Generation of almost simple groups, J. Algebra 178 (1995), 194–223. MR 1358262. DOI 10.1006/jabr.1995.1345.Google Scholar
[17] Darafsheh, M. R., Orders of elements in the groups related to the general linear group, Finite Fields Appl. 11 (2005), 738–747. MR 2181417. DOI 10.1016/j.ffa.2004.12.003.Google Scholar
[18] Dixon, J. D., The probability of generating the symmetric group, Math. Z. 110 (1969), 199–205. MR 0251758.Google Scholar
[19] Fulman, J. and Guralnick, R., Conjugacy class properties of the extension of GL(n,q) generated by the inverse transpose involution, J. Algebra 275 (2004), 356–396. MR 2047453. DOI 10.1016/j.jalgebra.2003.07.004.Google Scholar
[20] Gorenstein, D. and Lyons, R., The local structure of finite groups of characteristic 2 type, Mem. Amer. Math. Soc. 42 (1983), no. 276. MR 0690900.Google Scholar
[21] Gorenstein, D., Lyons, R., and Solomon, R., The Classification of the Finite Simple Groups, Math. Surveys Monogr. 40, Amer. Math. Soc., Providence, 1994. MR 1303592.CrossRefGoogle Scholar
[22] Guralnick, R. M., The spread of finite groups, in preparation.Google Scholar
[23] Guralnick, R. M. and Kantor, W. M., Probabilistic generation of finite simple groups, J. Algebra 234 (2000), 743–792. MR 1800754. DOI 10.1006/jabr.2000.8357.CrossRefGoogle Scholar
[24] Guralnick, R. M. and Malle, G., Products of conjugacy classes and fixed point spaces, J. Amer. Math. Soc. 25 (2012), 77–121. MR 2833479. DOI 10.1090/S0894-0347-2011-00709-1.CrossRefGoogle Scholar
[25] Guralnick, R., Pentilla, T., Praeger, C. E., and Saxl, J., Linear groups with orders having certain large prime divisors, Proc. Lond. Math. Soc. (3) 78 (1999), 167–214. MR 1658168. DOI 10.1112/S0024611599001616.Google Scholar
[26] Guralnick, R. M. and Saxl, J., Generation of finite almost simple groups by conjugates, J. Algebra 268 (2003), 519–571. MR 2009321. DOI 10.1016/S0021-8693(03)00182-0.Google Scholar
[27] Guralnick, R. M. and Shalev, A., On the spread of finite simple groups, Combinatorica 23 (2003), 73–87. MR 1996627. DOI 10.1007/s00493-003-0014-3.Google Scholar
[28] Hiss, G. and Malle, G., Low-dimensional representations of quasi-simple groups, LMS J. Comput. Math. 4 (2001), 22–63. MR 1835851.Google Scholar
[29] Jansen, C., Lux, K., Parker, R., and Wilson, R., An Atlas of Brauer Characters, London Math. Soc. Monogr. Ser. (N.S.) 11 Oxford University Press, New York, 1995. MR 1367961.Google Scholar
[30] Kantor, W. M., Subgroups of classical groups generated by long root elements, Trans. Amer. Math. Soc. 248, no. 2 (1979), 347–379. MR 0522265. DOI 10.2307/1998972.Google Scholar
[31] Kantor, W. M. and Lubotzky, A., The probability of generating a finite classical group, Geom. Dedicata 36 (1990), 67–87. MR 1065213. DOI 10.1007/BF00181465.CrossRefGoogle Scholar
[32] Kawanaka, N., On the irreducible characters of the finite unitary groups, J. Math. Soc. Japan 29 (1977), 425–450. MR 0450383.Google Scholar
[33] Kleidman, P. and Liebeck, M., The Subgroup Structure of the Finite Classical Groups, London Math. Soc. Lecture Note Ser. 129, Cambridge University Press, Cambridge, 1990. MR 1057341. DOI 10.1017/CBO9780511629235.Google Scholar
[34] Liebeck, M. W., The classification of finite simple Moufang loops, Math. Proc. Cambridge Philos. Soc. 102 (1987), 33–47. MR 0886433. DOI 10.1017/S0305004100067025.Google Scholar
[35] Liebeck, M. W. and Saxl, J., Minimal degrees of primitive permutation groups, with an application to monodromy groups of covers of Riemann surfaces, Proc. Lond. Math. Soc. (3) 63 (1991), 266–314. MR 1114511. DOI 10.1112/plms/s3-63.2.266.Google Scholar
[36] Liebeck, M. W. and Shalev, A., The probability of generating a finite simple group, Geom. Dedicata 56 (1995), 103–113. MR 1338320. DOI 10.1007/BF01263616.CrossRefGoogle Scholar
[37] Liebeck, M. W. and Shalev, A., Classical groups, probabilistic methods, and the (2,3)-generation problem, Ann. of Math. (2) 144 (1996), 77–125. MR 1405944. DOI 10.2307/2118584.Google Scholar
[38] Liebeck, M. W. and Shalev, A., Simple groups, permutation groups, and probability, J. Amer. Math. Soc. 12 (1999), 497–520. MR 1639620. DOI 10.1090/S0894-0347-99-00288-X.Google Scholar
[39] Liebeck, M. W. and Shalev, A., Random (r, s)-generation of finite classical groups, Bull. Lond. Math. Soc. 34 (2002), 185–188. MR 1874245. DOI 10.1112/S0024609301008827.Google Scholar
[40] Lübeck, F. and Malle, G., (2,3)-generation of exceptional groups, J. Lond. Math. Soc. (2) 59 (1999), 109–122. MR 1688493. DOI 10.1112/S002461079800670X.Google Scholar
[41] Lucchini, A. and Menegazzo, F., Generators for finite groups with a unique minimal normal subgroup, Rend. Semin. Mat. Univ. Padova 98 (1997), 173–191. MR 1492976.Google Scholar
[42] Ramanujan, S., “A proof of Bertrand’s postulate” in Collected Papers of Srinivasa Ramanujan, AMS Chelsea, Providence, 2000, 208–209. MR 2280867.Google Scholar
[43] Shalev, A., Random generation of finite simple groups by p-regular or p-singular elements, Israel J. Math. 125 (2001), 53–60. MR 1853805. DOI 10.1007/BF02773374.Google Scholar
[44] Shinoda, K., The characters of the finite conformal symplectic group, CSp(4,q), Comm. Algebra 10 (1982), 1369–1419. MR 0662708. DOI 10.1080/00927878208822782.Google Scholar
[45] Steinberg, R., Generators for simple groups, Canad. J. Math. 14 (1962), 277–283. MR 0143801.Google Scholar
[46] Steinberg, R., Lectures on Chevalley Groups, Yale University Press, New Haven, 1968. MR 0466335.Google Scholar
[47] Wall, G. E., On the conjugacy classes in the unitary, symplectic and orthogonal groups, J. Aust. Math. Soc. 3 (1963), 1–62. MR 0150210.CrossRefGoogle Scholar
[48] Zsigmondy, K., Zur Theorie der Potenzreste, Monatsh. Math. Phys. 3 (1892), 265–284. MR 1546236. DOI 10.1007/BF01692444.Google Scholar