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Every coprime linear group admits a base of size two


Authors: Zoltán Halasi and Károly Podoski
Journal: Trans. Amer. Math. Soc. 368 (2016), 5857-5887
MSC (2010): Primary 20C15; Secondary 20B99
DOI: https://doi.org/10.1090/tran/6544
Published electronically: December 15, 2015
MathSciNet review: 3458401
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Abstract: Let $ G$ be a linear group acting faithfully on a finite vector space $ V$ and assume that $ (\vert G\vert,\vert V\vert) =1$. In this paper we prove that $ G$ admits a base of size two and that this estimate is sharp. This generalizes and strengthens several former results concerning base sizes of coprime linear groups. As a direct consequence, we answer a question of I. M. Isaacs in the affirmative.


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  • [1] M. Aschbacher, Finite group theory, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 10, Cambridge University Press, Cambridge, 2000. MR 1777008 (2001c:20001)
  • [2] C. Benbenishty, On actions of primitive groups, PhD thesis, Hebrew University, Jerusalem, 2005.
  • [3] Timothy C. Burness, On base sizes for actions of finite classical groups, J. Lond. Math. Soc. (2) 75 (2007), no. 3, 545-562. MR 2352720 (2008m:20002), https://doi.org/10.1112/jlms/jdm033
  • [4] Timothy C. Burness, Robert M. Guralnick, and Jan Saxl, On base sizes for symmetric groups, Bull. Lond. Math. Soc. 43 (2011), no. 2, 386-391. MR 2781219 (2012d:20003), https://doi.org/10.1112/blms/bdq123
  • [5] Timothy C. Burness, Martin W. Liebeck, and Aner Shalev, Base sizes for simple groups and a conjecture of Cameron, Proc. Lond. Math. Soc. (3) 98 (2009), no. 1, 116-162. MR 2472163 (2009m:20001), https://doi.org/10.1112/plms/pdn024
  • [6] Timothy C. Burness, E. A. O'Brien, and Robert A. Wilson, Base sizes for sporadic simple groups, Israel J. Math. 177 (2010), 307-333. MR 2684423 (2011j:20035), https://doi.org/10.1007/s11856-010-0048-3
  • [7] Timothy C. Burness and Ákos Seress, On Pyber's base size conjecture, Trans. Amer. Math. Soc. 367 (2015), no. 8, 5633-5651. MR 3347185, https://doi.org/10.1090/S0002-9947-2015-06224-2
  • [8] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of finite groups, Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray, Oxford University Press, Eynsham, 1985. MR 827219 (88g:20025)
  • [9] Silvio Dolfi, Orbits of permutation groups on the power set, Arch. Math. (Basel) 75 (2000), no. 5, 321-327. MR 1785438 (2001g:20002), https://doi.org/10.1007/s000130050510
  • [10] Silvio Dolfi, Large orbits in coprime actions of solvable groups, Trans. Amer. Math. Soc. 360 (2008), no. 1, 135-152 (electronic). MR 2341997 (2008i:20027), https://doi.org/10.1090/S0002-9947-07-04155-4
  • [11] Joanna B. Fawcett, The base size of a primitive diagonal group, J. Algebra 375 (2013), 302-321. MR 2998958, https://doi.org/10.1016/j.jalgebra.2012.11.020
  • [12] The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.4.10; 2007. (http://www.gap-system.org)
  • [13] David Gluck, Trivial set-stabilizers in finite permutation groups, Canad. J. Math. 35 (1983), no. 1, 59-67. MR 685817 (84c:20008), https://doi.org/10.4153/CJM-1983-005-2
  • [14] David Gluck and Kay Magaard, Base sizes and regular orbits for coprime affine permutation groups, J. London Math. Soc. (2) 58 (1998), no. 3, 603-618. MR 1678153 (2000a:20006), https://doi.org/10.1112/S0024610798006802
  • [15] Dominic P. M. Goodwin, Regular orbits of linear groups with an application to the $ k(GV)$-problem. I, II, J. Algebra 227 (2000), no. 2, 395-432, 433-473. MR 1759829 (2001b:20010), https://doi.org/10.1006/jabr.1998.8078
  • [16] Z. Halasi and A. Maróti, The minimal base size for a $ p$-solvable linear group, arXiv:1310.5454 (2013)
  • [17] Brian Hartley and Alexandre Turull, On characters of coprime operator groups and the Glauberman character correspondence, J. Reine Angew. Math. 451 (1994), 175-219. MR 1277300 (95d:20010)
  • [18] B. Huppert, Endliche Gruppen. I, Die Grundlehren der Mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR 0224703 (37 #302)
  • [19] I. Martin Isaacs, Character theory of finite groups, Dover Publications, Inc., New York, 1994. Corrected reprint of the 1976 original [Academic Press, New York; MR0460423 (57 #417)]. MR 1280461
  • [20] I. M. Isaacs, Large orbits in actions of nilpotent groups, Proc. Amer. Math. Soc. 127 (1999), no. 1, 45-50. MR 1469413 (99b:20035), https://doi.org/10.1090/S0002-9939-99-04584-0
  • [21] 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 (91g:20001)
  • [22] Christoph Köhler and Herbert Pahlings, Regular orbits and the $ k(GV)$-problem, Groups and computation, III (Columbus, OH, 1999) Ohio State Univ. Math. Res. Inst. Publ., vol. 8, de Gruyter, Berlin, 2001, pp. 209-228. MR 1829482 (2002d:20010)
  • [23] Martin W. Liebeck and Aner Shalev, Simple groups, permutation groups, and probability, J. Amer. Math. Soc. 12 (1999), no. 2, 497-520. MR 1639620 (99h:20004), https://doi.org/10.1090/S0894-0347-99-00288-X
  • [24] Martin W. Liebeck and Aner Shalev, Bases of primitive linear groups, J. Algebra 252 (2002), no. 1, 95-113. MR 1922387 (2003f:20005), https://doi.org/10.1016/S0021-8693(02)00001-7
  • [25] Gunter Malle and Gabriel Navarro, Blocks with equal height zero degrees, Trans. Amer. Math. Soc. 363 (2011), no. 12, 6647-6669. MR 2833571 (2012g:20016), https://doi.org/10.1090/S0002-9947-2011-05333-X
  • [26] Gunter Malle and Donna Testerman, Linear algebraic groups and finite groups of Lie type, Cambridge Studies in Advanced Mathematics, vol. 133, Cambridge University Press, Cambridge, 2011. MR 2850737 (2012i:20058)
  • [27] Alexander Moretó and Thomas R. Wolf, Orbit sizes, character degrees and Sylow subgroups, Adv. Math. 184 (2004), no. 1, 18-36. MR 2047847 (2005b:20015a), https://doi.org/10.1016/S0001-8708(03)00093-8
  • [28] D. S. Passman, Groups with normal solvable Hall $ p^{\prime } $-subgroups, Trans. Amer. Math. Soc. 123 (1966), 99-111. MR 0195947 (33 #4143)
  • [29] P. P. Pálfy and L. Pyber, Small groups of automorphisms, Bull. London Math. Soc. 30 (1998), no. 4, 386-390. MR 1620821 (99g:20040), https://doi.org/10.1112/S0024609397004037
  • [30] László Pyber, Asymptotic results for permutation groups, Groups and computation (New Brunswick, NJ, 1991) DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 11, Amer. Math. Soc., Providence, RI, 1993, pp. 197-219. MR 1235804 (94g:20003)
  • [31] Ákos Seress, The minimal base size of primitive solvable permutation groups, J. London Math. Soc. (2) 53 (1996), no. 2, 243-255. MR 1373058 (96k:20003), https://doi.org/10.1112/jlms/53.2.243
  • [32] Ákos Seress, Primitive groups with no regular orbits on the set of subsets, Bull. London Math. Soc. 29 (1997), no. 6, 697-704. MR 1468057 (98g:20008), https://doi.org/10.1112/S0024609397003536
  • [33] Ákos Seress, Permutation group algorithms, Cambridge Tracts in Mathematics, vol. 152, Cambridge University Press, Cambridge, 2003. MR 1970241 (2004c:20008)
  • [34] E. P. Vdovin, Regular orbits of solvable linear $ p'$-groups, Sib. Èlektron. Mat. Izv. 4 (2007), 345-360. MR 2465432 (2009k:20114)
  • [35] Thomas R. Wolf, Large orbits of supersolvable linear groups, J. Algebra 215 (1999), no. 1, 235-247. MR 1684166 (2000d:20047), https://doi.org/10.1006/jabr.1998.7730

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Additional Information

Zoltán Halasi
Affiliation: Institute of Mathematics, University of Debrecen, P.O. Box 12, H-4010 Debrecen, Hungary
Address at time of publication: Department of Algebra and Number Theory, Eötvös University, Pázmány Péter sétány 1/c, 1117 Budapest, Hungary
Email: halasi.zoltan@renyi.mta.hu

Károly Podoski
Affiliation: Budapest Business School, College of Finance and Accountancy, Buzogány Street 10-12, H-1149 Budapest, Hungary
Address at time of publication: Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13-15, 1053 Budapest, Hungary
Email: podoski.karoly@pszfb.bgf.hu, podoski.karoly@renyi.mta.hu

DOI: https://doi.org/10.1090/tran/6544
Keywords: Coprime linear group, base size, regular partition
Received by editor(s): December 26, 2013
Received by editor(s) in revised form: June 30, 2014, and August 3, 2014
Published electronically: December 15, 2015
Additional Notes: The research leading to these results has received funding from the European Union’s Seventh Framework Programme (FP7/2007-2013) under grant agreement no. 318202, from MTA Rényi Institute Lendület Limits of Structures Research Group and from OTKA K84233.
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