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SIMPLE GROUPS, PRODUCT ACTIONS, AND GENERALIZED QUADRANGLES

  • JOHN BAMBERG (a1), TOMASZ POPIEL (a2) (a3) and CHERYL E. PRAEGER (a2) (a4)

Abstract

The classification of flag-transitive generalized quadrangles is a long-standing open problem at the interface of finite geometry and permutation group theory. Given that all known flag-transitive generalized quadrangles are also point-primitive (up to point–line duality), it is likewise natural to seek a classification of the point-primitive examples. Working toward this aim, we are led to investigate generalized quadrangles that admit a collineation group $G$ preserving a Cartesian product decomposition of the set of points. It is shown that, under a generic assumption on $G$ , the number of factors of such a Cartesian product can be at most four. This result is then used to treat various types of primitive and quasiprimitive point actions. In particular, it is shown that $G$ cannot have holomorph compound O’Nan–Scott type. Our arguments also pose purely group-theoretic questions about conjugacy classes in nonabelian finite simple groups and fixities of primitive permutation groups.

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John Bamberg is supported by the Australian Research Council Future Fellowship FT120100036. Tomasz Popiel and Cheryl E. Praeger are supported by the Australian Research Council Discovery Grant DP140100416.

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[1] Babai, L., On the automorphism groups of strongly regular graphs II , J. Algebra 421 (2015), 560578.10.1016/j.jalgebra.2014.09.007
[2] Baddeley, R. W., Primitive permutation groups with a regular non-abelian normal subgroup , Proc. Lond. Math. Soc. 67 (1993), 547595.
[3] Bamberg, J., Betten, A., Cara, P., De Beule, J., Lavrauw, M. and Neunhöffer, M., FinInG–Finite Incidence Geometry, Version 1.0, 2014.
[4] Bamberg, J. and Giudici, M., Point regular groups of automorphisms of generalised quadrangles , J. Combin. Theory Ser. A 118 (2011), 11141128.10.1016/j.jcta.2010.11.004
[5] Bamberg, J., Giudici, M., Morris, J., Royle, G. F. and Spiga, P. , Generalised quadrangles with a group of automorphisms acting primitively on points and lines , J. Combin. Theory Ser. A 119 (2012), 14791499.10.1016/j.jcta.2012.04.005
[6] Bamberg, J., Glasby, S. P., Popiel, T. and Praeger, C. E., Generalised quadrangles and transitive pseudo-hyperovals , J. Combin. Des. 24 (2016), 151164.10.1002/jcd.21411
[7] Bamberg, J., Glasby, S. P., Popiel, T., Praeger, C. E. and Schneider, C., Point-primitive generalised hexagons and octagons , J. Combin. Theory Ser. A 147 (2017), 186204.
[8] Bamberg, J., Popiel, T. and Praeger, C. E., Point-primitive, line-transitive generalised quadrangles of holomorph type , J. Group Theory 20 (2017), 269287.
[9] Burness, T. C. and Giudici, M., Classical Groups, Derangements, and Primes, Australian Mathematical Society Lecture Series 25 , Cambridge University Press, Cambridge, 2016.
[10] Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A., ATLAS of Finite Groups, Oxford University Press, Oxford, 1985.
[11] Covato, E., The involution fixity of simple groups, Ph.D. thesis, University of Bristol, 2017.
[12] De Winter, S. and Thas, K., Generalized quadrangles with an abelian Singer group , Des. Codes Cryptogr. 39 (2006), 8187.10.1007/s10623-005-2747-z
[13] Feit, W. and Thompson, J. G., Solvability of groups of odd order , Pacific J. Math. 13 (1963), 7751029.10.2140/pjm.1963.13.775
[14] Frohardt, D. and Magaard, K., Fixed point ratios in exceptional groups of rank at most two , Comm. Algebra 30 (2002), 571602.
[15]The GAP Group, GAP–Groups, Algorithms, and Programming, Version 4.7.8, 2015.
[16] Ghinelli, D., Regular groups on generalized quadrangles and nonabelian difference sets with multiplier - 1 , Geom. Dedicata 41 (1992), 165174.
[17] Gorenstein, D., Lyons, R. and Solomon, R., The Classification of the Finite Simple Groups, Number 3, Mathematical Surveys and Monographs 40 , American Mathematical Society, Providence, 1998.
[18] Liebeck, M. W. and Seitz, G. M., Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras, Mathematical Surveys and Monographs 180 , American Mathematical Society, Providence, RI, 2012.10.1090/surv/180
[19] Liebeck, M. W. and Shalev, A., On fixed points of elements in primitive permutation groups , J. Algebra 421 (2015), 438459.10.1016/j.jalgebra.2014.08.038
[20] Morgan, L. and Popiel, T., Generalised polygons admitting a point-primitive almost simple group of Suzuki or Ree type , Electron. J. Combin. 23 (2016), P1.34.
[21] Payne, S. E. and Thas, J. A., Finite Generalized Quadrangles, Pitman, London, 1984.
[22] Praeger, C. E., Li, C.-H. and Niemeyer, A. C., “ Finite transitive permutation groups and finite vertex-transitive graphs ”, in Graph Symmetry, (eds. Hahn, G. and Sabidussi, G.) Kluwer, 1997, 277318.10.1007/978-94-015-8937-6_7
[23] Spaltenstein, N., Caractères unipotents de 3D4(F q ) , Comment. Math. Helv. 57 (1982), 676691.
[24] Suzuki, M., On a class of doubly transitive groups , Annals Math. 75 (1962), 105145.
[25] Tits, J., Sur la trialité et certains groupes qui s’en déduisent , Publ. Math. Inst. Hautes Études Sci. 2 (1959), 1360.
[26] Van Maldeghem, H., Generalized Polygons, Birkháuser, Basel, 1998.
[27] Yoshiara, S., A generalized quadrangle with an automorphism group acting regularly on the points , European J. Combin. 28 (2007), 653664.
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SIMPLE GROUPS, PRODUCT ACTIONS, AND GENERALIZED QUADRANGLES

  • JOHN BAMBERG (a1), TOMASZ POPIEL (a2) (a3) and CHERYL E. PRAEGER (a2) (a4)

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