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No sporadic almost simple group acts primitively on the points of a generalised quadrangle. (English) Zbl 07327833
Summary: A generalised quadrangle is a point-line incidence geometry $$\mathcal{G}$$ such that: (i) any two points lie on at most one line, and (ii) given a line $$L$$ and a point $$p$$ not incident with $$L$$, there is a unique point on $$L$$ collinear with $$p$$. They are a specific case of the generalised polygons introduced by J. Tits [Publ. Math., Inst. Hautes Étud. Sci. 2, 13–60 (1959; Zbl 0088.37204)], and these structures and their automorphism groups are of some importance in finite geometry. An integral part of understanding the automorphism groups of finite generalised quadrangles is knowing which groups can act primitively on their points, and in particular, which almost simple groups arise as automorphism groups. We show that no sporadic almost simple group can act primitively on the points of a finite (thick) generalised quadrangle. We also present two new ideas contributing towards analysing point-primitive groups acting on generalised quadrangles. The first is the outline and implementation of an algorithm for determining whether a given group can act primitively on the points of some generalised quadrangle. The second is the discussion of a conjecture resulting from observations made in the course of this work: any group acting primitively on the points of a generalised quadrangle must either act transitively on lines or have exactly two line-orbits, each containing half of the lines.
##### MSC:
 20B15 Primitive groups 20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
##### Software:
AtlasRep; DESIGN; FinInG; GAP; GRAPE
Full Text:
##### References:
 [1] Bamberg, J.; Betten, A.; Cara, P.; De Beule, J.; Lavrauw, M.; Neunhöffer, M., FinInG - Finite incidence geometry (2018), Version 1.4.1 [2] Bamberg, J.; Giudici, M.; Morris, J.; Royle, G. F.; Spiga, P., Generalised quadrangles with a group of automorphisms acting primitively on points and lines, J. Combin. Theory Ser. A, 119, 7, 1479-1499 (2012) · Zbl 1245.05014 [3] Bamberg, J.; Glasby, S.; Popiel, T.; Praeger, C., Generalized quadrangles and transitive pseudo-hyperovals, J. Combin. Des., 24, 4, 151-164 (2016), 3 · Zbl 1338.05031 [4] Bamberg, J.; Popiel, T.; Praeger, C. E., Point-primitive, line-transitive generalised quadrangles of holomorph type, J. Group Theory, 20, 2, 269-287 (2017) · Zbl 1428.20004 [5] Bamberg, J.; Popiel, T.; Praeger, C. E., Simple groups, product actions, and generalized quadrangles, Nagoya Math. J., 1-40 (2017) [6] Bray, J. N.; Holt, D. F.; Roney-Dougal, C. M., The maximal subgroups of the low-dimensional finite classical groups, (London Mathematical Society Lecture Note Series, vol. 407 (2013), Cambridge University Press: Cambridge University Press Cambridge), With a foreword by Martin Liebeck · Zbl 1303.20053 [7] Brouwer, A. E.; Cohen, A. M.; Neumaier, A., (Distance-Regular Graphs. Distance-Regular Graphs, A Series of Modern Surveys in Mathematics (1989), Springer-Verlag Berlin Heidelberg) · Zbl 0747.05073 [8] Buekenhout, F.; Van Maldeghem, H., Remarks on finite generalized hexagons and octagons with a point-transitive automorphism group, (Finite Geometry and Combinatorics (Deinze, 1992). Finite Geometry and Combinatorics (Deinze, 1992), London Math. Soc. Lecture Note Ser., vol. 191 (1992), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 89-102 · Zbl 0792.51004 [9] Buekenhout, F.; Van Maldeghem, H., Finite distance-transitive generalized polygons, Geom. Dedicata, 52, 1, 41-51 (1994) · Zbl 0809.51008 [10] Cameron, P. J., Finite geometry after Aschbacher’s theorem: $$\operatorname{PGL} ( n , q )$$ from a Kleinian viewpoint, (Geometry, Combinatorial Designs and Related Structures (Spetses, 1996). Geometry, Combinatorial Designs and Related Structures (Spetses, 1996), London Math. Soc. Lecture Note Ser., vol. 245 (1996), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 43-61 · Zbl 0885.20019 [11] Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A., Atlas of finite groups, (Maximal Subgroups and Ordinary Characters for Simple Groups (1985), Oxford University Press: Oxford University Press Eynsham), With computational assistance from J. G. Thackray · Zbl 0568.20001 [12] Dembowski, P., Finite geometries, (Classics in Mathematics (1997), Springer-Verlag: Springer-Verlag Berlin), Reprint of the 1968 original · Zbl 0159.50001 [13] Feit, W.; Higman, G., The nonexistence of certain generalized polygons, J. Algebra, 1, 114-131 (1964) · Zbl 0126.05303 [14] Fong, P.; Seitz, G. M., Groups with a (b, n)-pair of rank 2, (Gagen, T.; Hale, M. P.; Shult, E. E., Finite Groups ’72. Finite Groups ’72, North-Holland Mathematics Studies, vol. 7 (1973), North-Holland), 36-40 [15] Giudici, M.; Glasby, S. P.; Praeger, C. E., Subgroups of classical groups that are transitive on subspaces (2020), https://arxiv.org/abs/2012.07213 [16] Kantor, W. M., Flag-transitive planes, (Finite Geometries (Winnipeg, Man., 1984). Finite Geometries (Winnipeg, Man., 1984), Lecture Notes in Pure and Appl. Math., vol. 103 (1984), Dekker: Dekker New York), 179-181 [17] Kleidman, P.; Liebeck, M. W., The Subgroup Structure of the Finite Classical Groups (1990), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0697.20004 [18] Payne, S. E.; Thas, J. A., Finite Generalised Quadrangles (2009), European Mathematical Society [19] Praeger, C. E., The inclusion problem for finite primitive permutation groups, Proc. Lond. Math. Soc., s3-60, 1, 68-88 (1990) · Zbl 0653.20005 [20] Schneider, C.; Van Maldeghem, H., Primitive flag-transitive generalized hexagons and octagons, J. Combin. Theory Ser. A, 115, 8, 1436-1455 (2008) · Zbl 1202.20003 [21] Segre, B., Forme e geometrie hermitiane con particolare rigwrdo al case finito, Ann. Mat. Pura Appl., 4, 70, 1-202 (1965) · Zbl 0146.16703 [22] Soicher, L. H., The DESIGN package for GAP (2011), Version 1.6 [23] Soicher, L. H., The GRAPE package for GAP (2018), Version 4.8.1 [24] Thas, J., Interesting pointsets in generalized quadrangles and partial geometries, Linear Algebra Appl., 114-115, 103-131 (1989), (special issue dedicated to Alan J. Hoffman) · Zbl 0663.05014 [25] The GAP Group, J., GAP - Groups, algorithms, and programming (2018), Version 4.10.0 [26] Tits, J., Sur la trialité et certains groupes qui s’en déduisent, Publ. Math. Inst. Hautes Études Sci., 2, 13-60 (1959) · Zbl 0088.37204 [27] Tits, J., Buildings of spherical type and finite bn-pairs, (Lecture Notes in Mathematics, vol. 386 (1974), Springer-Verlag Berlin Heidelberg) · Zbl 0295.20047 [28] Van Maldeghem, H., Generalized polygons, (Modern Birkhäuser Classics (1998), Birkhäuser/Springer Basel AG: Birkhäuser/Springer Basel AG Basel), [2011 reprint of the 1998 original] [MR1725957] · Zbl 0914.51005 [29] Wilson, R. A., Maximal subgroups of sporadic groups (2017), arXiv e-prints, arXiv:1701.02095 · Zbl 1448.20018 [30] Wilson, R. A.; Parker, R. A.; Nickerson, S.; Bray, J. N.; Breuer, T., AtlasRep, a GAP interface to the atlas of group representations (2016), GAP package
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