Arithmetic results on orbits of linear groups

Giudici, Michael; Liebeck, Martin; Praeger, Cheryl; Saxl, Jan; Tiep, Pham

doi:10.1090/tran/6373

Arithmetic results on orbits of linear groups
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by Michael Giudici, Martin W. Liebeck, Cheryl E. Praeger, Jan Saxl and Pham Huu Tiep PDF

Trans. Amer. Math. Soc. 368 (2016), 2415-2467 Request permission

Abstract:

Let $p$ be a prime and $G$ a subgroup of $GL_d(p)$. We define $G$ to be $p$-exceptional if it has order divisible by $p$, but all its orbits on vectors have size coprime to $p$. We obtain a classification of $p$-exceptional linear groups. This has consequences for a well-known conjecture in representation theory, and also for a longstanding question concerning $\frac {1}{2}$-transitive linear groups (i.e. those having all orbits on nonzero vectors of equal length), classifying those of order divisible by $p$.

References

M. Aschbacher, On the maximal subgroups of the finite classical groups, Invent. Math. 76 (1984), no. 3, 469–514. MR 746539, DOI 10.1007/BF01388470
John Bamberg, Michael Giudici, Martin W. Liebeck, Cheryl E. Praeger, and Jan Saxl, The classification of almost simple $\frac 32$-transitive groups, Trans. Amer. Math. Soc. 365 (2013), no. 8, 4257–4311. MR 3055696, DOI 10.1090/S0002-9947-2013-05758-3
J. van Bon, A. A. Ivanov, and J. Saxl, Affine distance-transitive graphs with sporadic stabilizer, European J. Combin. 20 (1999), no. 2, 163–177. MR 1676190, DOI 10.1006/eujc.1998.0268
Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, DOI 10.1006/jsco.1996.0125
Peter J. Cameron, Peter M. Neumann, and Jan Saxl, On groups with no regular orbits on the set of subsets, Arch. Math. (Basel) 43 (1984), no. 4, 295–296. MR 802301, DOI 10.1007/BF01196649
Roger W. Carter, Finite groups of Lie type, Wiley Classics Library, John Wiley & Sons, Ltd., Chichester, 1993. Conjugacy classes and complex characters; Reprint of the 1985 original; A Wiley-Interscience Publication. MR 1266626
Arjeh M. Cohen, Kay Magaard, and Sergey Shpectorov, Affine distance-transitive graphs: the cross characteristic case, European J. Combin. 20 (1999), no. 5, 351–373. MR 1702374, DOI 10.1006/eujc.1999.0286
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, $\Bbb {ATLAS}$ of finite groups, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. MR 827219
C. W. Curtis, Modular representations of finite groups with split $(B,\,N)$-pairs, Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69) Springer, Berlin, 1970, pp. 57–95. MR 0262383
Leonard Eugene Dickson, Linear groups: With an exposition of the Galois field theory, Dover Publications, Inc., New York, 1958. With an introduction by W. Magnus. MR 0104735
Silvio Dolfi, Robert Guralnick, Cheryl E. Praeger, and Pablo Spiga, Coprime subdegrees for primitive permutation groups and completely reducible linear groups, Israel J. Math. 195 (2013), no. 2, 745–772. MR 3096572, DOI 10.1007/s11856-012-0163-4
Larry Dornhoff, Group representation theory. Part B: Modular representation theory, Pure and Applied Mathematics, vol. 7, Marcel Dekker, Inc., New York, 1972. MR 0347960
Walter Feit, The representation theory of finite groups, North-Holland Mathematical Library, vol. 25, North-Holland Publishing Co., Amsterdam-New York, 1982. MR 661045
A. Gambini Weigel and T. S. Weigel, On the orders of primitive linear $p’$-groups, Bull. Austral. Math. Soc. 48 (1993), no. 3, 495–521. MR 1248053, DOI 10.1017/S0004972700015951
The GAP group, ‘GAP - groups, algorithms, and programming’, Version 4.4, 2004, http://www.gap-system.org.
David Gluck, Kay Magaard, Udo Riese, and Peter Schmid, The solution of the $k(GV)$-problem, J. Algebra 279 (2004), no. 2, 694–719. MR 2078936, DOI 10.1016/j.jalgebra.2004.02.027
David Gluck and Thomas R. Wolf, Defect groups and character heights in blocks of solvable groups. II, J. Algebra 87 (1984), no. 1, 222–246. MR 736777, DOI 10.1016/0021-8693(84)90168-6
David Gluck and Thomas R. Wolf, Brauer’s height conjecture for $p$-solvable groups, Trans. Amer. Math. Soc. 282 (1984), no. 1, 137–152. MR 728707, DOI 10.1090/S0002-9947-1984-0728707-2
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, DOI 10.1006/jabr.1998.8078
Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups. Number 3. Part I. Chapter A, Mathematical Surveys and Monographs, vol. 40, American Mathematical Society, Providence, RI, 1998. Almost simple $K$-groups. MR 1490581, DOI 10.1090/surv/040.3
Robert M. Guralnick, Generation of simple groups, J. Algebra 103 (1986), no. 1, 381–401. MR 860714, DOI 10.1016/0021-8693(86)90194-8
Robert M. Guralnick and Jan Saxl, Generation of finite almost simple groups by conjugates, J. Algebra 268 (2003), no. 2, 519–571. MR 2009321, DOI 10.1016/S0021-8693(03)00182-0
Christoph Hering, Transitive linear groups and linear groups which contain irreducible subgroups of prime order. II, J. Algebra 93 (1985), no. 1, 151–164. MR 780488, DOI 10.1016/0021-8693(85)90179-6
Gerhard Hiss and Gunter Malle, Corrigenda: “Low-dimensional representations of quasi-simple groups” [LMS J. Comput. Math. 4 (2001), 22–63; MR1835851 (2002b:20015)], LMS J. Comput. Math. 5 (2002), 95–126. MR 1942256, DOI 10.1112/S1461157000000711
G. D. James, On the minimal dimensions of irreducible representations of symmetric groups, Math. Proc. Cambridge Philos. Soc. 94 (1983), no. 3, 417–424. MR 720791, DOI 10.1017/S0305004100000803
Christoph Jansen, Klaus Lux, Richard Parker, and Robert Wilson, An atlas of Brauer characters, London Mathematical Society Monographs. New Series, vol. 11, The Clarendon Press, Oxford University Press, New York, 1995. Appendix 2 by T. Breuer and S. Norton; Oxford Science Publications. MR 1367961
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, DOI 10.1017/CBO9780511629235
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
Vicente Landazuri and Gary M. Seitz, On the minimal degrees of projective representations of the finite Chevalley groups, J. Algebra 32 (1974), 418–443. MR 360852, DOI 10.1016/0021-8693(74)90150-1
Martin W. Liebeck, The affine permutation groups of rank three, Proc. London Math. Soc. (3) 54 (1987), no. 3, 477–516. MR 879395, DOI 10.1112/plms/s3-54.3.477
Martin W. Liebeck, Cheryl E. Praeger, and Jan Saxl, Regular subgroups of primitive permutation groups, Mem. Amer. Math. Soc. 203 (2010), no. 952, vi+74. MR 2588738, DOI 10.1090/S0065-9266-09-00569-9
Martin W. Liebeck and Jan Saxl, The primitive permutation groups of odd degree, J. London Math. Soc. (2) 31 (1985), no. 2, 250–264. MR 809946, DOI 10.1112/jlms/s2-31.2.250
Gunter Malle, Fast-einfache Gruppen mit langen Bahnen in absolut irreduzibler Operation, J. Algebra 300 (2006), no. 2, 655–672 (German, with German summary). MR 2228215, DOI 10.1016/j.jalgebra.2006.01.012
Masafumi Murai, On Brauer’s height zero conjecture, Proc. Japan Acad. Ser. A Math. Sci. 88 (2012), no. 3, 38–40. MR 2908621
G. Navarro and B. Späth, A reduction theorem for the Brauer height zero conjecture, J. Eur. Math. Soc. (to appear).
Gabriel Navarro and Pham Huu Tiep, Brauer’s height zero conjecture for the 2-blocks of maximal defect, J. Reine Angew. Math. 669 (2012), 225–247. MR 2980457, DOI 10.1515/crelle.2011.147
Gabriel Navarro and Pham Huu Tiep, Characters of relative $p’$-degree over normal subgroups, Ann. of Math. (2) 178 (2013), no. 3, 1135–1171. MR 3092477, DOI 10.4007/annals.2013.178.3.7
D. S. Passman, Solvable ${3\over 2}$-transitive permutation groups, J. Algebra 7 (1967), 192–207. MR 235018, DOI 10.1016/0021-8693(67)90055-5
D. S. Passman, Exceptional $3/2$-transitive permutation groups, Pacific J. Math. 29 (1969), 669–713. MR 244363
Gary M. Seitz and Alexander E. Zalesskii, On the minimal degrees of projective representations of the finite Chevalley groups. II, J. Algebra 158 (1993), no. 1, 233–243. MR 1223676, DOI 10.1006/jabr.1993.1132
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, DOI 10.1112/S0024609397003536
Pham Huu Tiep and A. E. Zalesskiĭ, Unipotent elements of finite groups of Lie type and realization fields of their complex representations, J. Algebra 271 (2004), no. 1, 327–390. MR 2022486, DOI 10.1016/S0021-8693(03)00174-1
Ascher Wagner, An observation on the degrees of projective representations of the symmetric and alternating group over an arbitrary field, Arch. Math. (Basel) 29 (1977), no. 6, 583–589. MR 460451, DOI 10.1007/BF01220457
Harold N. Ward, Representations of symplectic groups, J. Algebra 20 (1972), 182–195. MR 286909, DOI 10.1016/0021-8693(72)90098-1
Helmut Wielandt, Finite permutation groups, Academic Press, New York-London, 1964. Translated from the German by R. Bercov. MR 0183775
R. A. Wilson, et al., A World-Wide-Web Atlas of finite group representations, http://brauer.maths.qmul.ac.uk/Atlas/v3/.
Hans Zassenhaus, Über endliche Fastkörper, Abh. Math. Sem. Univ. Hamburg 11 (1935), no. 1, 187–220 (German). MR 3069653, DOI 10.1007/BF02940723