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A classification of finite locally 2-transitive generalized quadrangles. (English) Zbl 07313190
A generalized quadrangle is an incidence geometry of points and lines such that every pair of distinct points determines at most one line and every line contains at least two distinct points, satisfying the GQ Axiom: Given a point \(P\) and a line \(\ell\) not incident with \(P,\) there is a unique point on \(\ell\) collinear with \(P.\)
One of the outstanding open questions in the area is the classification of flag-transitive finite generalized quadrangles, that is, the classification of all finite generalized quadrangles with a group of collineations that is transitive on incident point-line pairs.
It has been conjectured by W.M. Kantor that the only non-classical finite flag-transitive generalized quadrangles are (up to duality) either the unique generalized quadrangle of order \((3, 5)\) or the generalized quadrangle of order \((15, 17)\) arising from the Lunelli-Sce hyperoval.
We recall that an antiflag is a non-incident point-line pair. Notice that, by the GQ Axiom, antiflag-transitivity implies flag-transitivity for a generalized quadrangle.
In [J. Bamberg et al., Trans. Amer. Math. Soc. 370, 1551–1601 (2018, Zbl 1381.51001)] the authors have proved that, up to duality, the only non-classical antiflag-transitive generalized quadrangle is the unique generalized quadrangle of order \((3, 5).\)
If \(G\) is a subgroup of collineations of a finite generalized quadrangle \(Q\) that is transitive both on pairs of collinear points and on pairs of concurrent lines, then \(Q\) is said to be a locally \((G, 2)-\)transitive generalized quadrangle. The authors prove the following theorem:
Theorem 1 If \(Q\) is a thick finite locally \((G, 2)\)-transitive generalized quadrangle and \(Q\) is not a classical generalized quadrangle, then (up to duality) \(Q\) is the unique generalized quadrangle of order \((3, 5).\)
An equivalent definition of locally \(2-\)transitive generalized quadrangle is that it has an incidence graph that is locally \(2\)-arc-transitive. Since an equivalent definition of an antiflag-transitive generalized quadrangle is that it has a locally \(3-\)arc-transitive incidence graph, Theorem 1 provides further progress towards Kantor conjecture.

MSC:
51E12 Generalized quadrangles and generalized polygons in finite geometry
20B05 General theory for finite permutation groups
20B15 Primitive groups
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
Software:
GAP; Magma
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[1] Alavi, Seyed Hassan; Burness, Timothy C., Large subgroups of simple groups, J. Algebra, 421, 187-233 (2015) · Zbl 1308.20012
[2] Aschbacher, Michael, \(S_3\)-free 2-fusion systems, Proc. Edinb. Math. Soc. (2), 56, 1, 27-48 (2013) · Zbl 1278.20021
[3] Bamberg, John; Giudici, Michael; Morris, Joy; Royle, Gordon F.; Spiga, Pablo, 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
[4] Bamberg, John; Glasby, S. P.; Popiel, Tomasz; Praeger, Cheryl E., Generalized quadrangles and transitive pseudo-hyperovals, J. Combin. Des., 24, 4, 151-164 (2016) · Zbl 1338.05031
[5] Bamberg, John; Li, Cai Heng; Swartz, Eric, A classification of finite antiflag-transitive generalized quadrangles, Trans. Amer. Math. Soc., 370, 3, 1551-1601 (2018) · Zbl 1381.51001
[6] Bamberg, John; Popiel, Tomasz; Praeger, Cheryl E., Point-primitive, line-transitive generalised quadrangles of holomorph type, J. Group Theory, 20, 2, 269-287 (2017) · Zbl 1428.20004
[7] Bamberg, John; Popiel, Tomasz; Praeger, Cheryl E., Simple groups, product actions, and generalized quadrangles, Nagoya Math. J., 234, 87-126 (2019) · Zbl 1431.51003
[8] Bichara, Alessandro; Mazzocca, Francesco; Somma, Clelia, On the classification of generalized quadrangles in a finite affine space \({\rm AG}(3,\,2^h)\), Boll. Un. Mat. Ital. B (5), 17, 1, 298-307 (1980) · Zbl 0463.51013
[9] Bosma, Wieb; Cannon, John; Playoust, Catherine, The Magma algebra system. I. The user language, J. Symbolic Comput., 24, 3-4, 235-265 (1997) · Zbl 0898.68039
[10] Bray, John N.; Holt, Derek F.; Roney-Dougal, Colva M., The maximal subgroups of the low-dimensional finite classical groups, London Mathematical Society Lecture Note Series 407, xiv+438 pp. (2013), Cambridge University Press, Cambridge · Zbl 1303.20053
[11] Brouwer, A. E.; Cohen, A. M.; Neumaier, A., Distance-regular graphs, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 18, xviii+495 pp. (1989), Springer-Verlag, Berlin · Zbl 0747.05073
[12] Brown, Julia M. N.; Cherowitzo, William E., The Lunelli-Sce hyperoval in \(\rm PG(2,16)\), J. Geom., 69, 1-2, 15-36 (2000) · Zbl 0968.51004
[13] Buekenhout, F.; Van Maldeghem, H., Finite distance-transitive generalized polygons, Geom. Dedicata, 52, 1, 41-51 (1994) · Zbl 0809.51008
[14] Cameron, Peter J., Permutation groups, London Mathematical Society Student Texts 45, x+220 pp. (1999), Cambridge University Press, Cambridge · Zbl 0922.20003
[15] Fan, Wenwen; Leemans, Dimitri; Li, Cai Heng; Pan, Jiangmin, Locally 2-arc-transitive complete bipartite graphs, J. Combin. Theory Ser. A, 120, 3, 683-699 (2013) · Zbl 1259.05121
[16] Fong, Paul; Seitz, Gary M., Groups with a \((B,\,N)\)-pair of rank \(2\). I, II, Invent. Math., 21, 1-57; ibid. 24 (1974), 191-239 (1973) · Zbl 0295.20048
[17] GAP4 The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.10.0, 2018.
[18] Giudici, Michael; Li, Cai Heng; Praeger, Cheryl E., Analysing finite locally \(s\)-arc transitive graphs, Trans. Amer. Math. Soc., 356, 1, 291-317 (2004) · Zbl 1022.05033
[19] Giudici, Michael; Li, Cai Heng; Praeger, Cheryl E., Characterizing finite locally \(s\)-arc transitive graphs with a star normal quotient, J. Group Theory, 9, 5, 641-658 (2006) · Zbl 1113.05047
[20] Giudici, Michael; Li, Cai Heng; Praeger, Cheryl E., Locally \(s\)-arc transitive graphs with two different quasiprimitive actions, J. Algebra, 299, 2, 863-890 (2006) · Zbl 1092.05028
[21] Gorenstein, Daniel; Lyons, Richard; Solomon, Ronald, The classification of the finite simple groups, Mathematical Surveys and Monographs 40, xiv+165 pp. (1994), American Mathematical Society, Providence, RI · Zbl 0816.20016
[22] Guralnick, Robert; Penttila, Tim; Praeger, Cheryl E.; Saxl, Jan, Linear groups with orders having certain large prime divisors, Proc. London Math. Soc. (3), 78, 1, 167-214 (1999) · Zbl 1041.20035
[23] Guralnick, Robert M.; Mar\'{o}ti, Attila; Pyber, L\'{a}szl\'{o}, Normalizers of primitive permutation groups, Adv. Math., 310, 1017-1063 (2017) · Zbl 1414.20002
[24] Kantor, W. M., Automorphism groups of some generalized quadrangles. Advances in finite geometries and designs, Chelwood Gate, 1990, Oxford Sci. Publ., 251-256 (1991), Oxford Univ. Press, New York · Zbl 0736.51002
[25] Kleidman, Peter; Liebeck, Martin, The subgroup structure of the finite classical groups, London Mathematical Society Lecture Note Series 129, x+303 pp. (1990), Cambridge University Press, Cambridge · Zbl 0697.20004
[26] Li, Cai Heng; Seress, \'{A}kos; Song, Shu Jiao, \(s\)-arc-transitive graphs and normal subgroups, J. Algebra, 421, 331-348 (2015) · Zbl 1301.05172
[27] Liebeck, Martin W.; Praeger, Cheryl E.; Saxl, Jan, The maximal factorizations of the finite simple groups and their automorphism groups, Mem. Amer. Math. Soc., 86, 432, iv+151 pp. (1990) · Zbl 0703.20021
[28] Morgan, Luke; Swartz, Eric; Verret, Gabriel, On 2-arc-transitive graphs of order \(kp^n\), J. Combin. Theory Ser. B, 117, 77-87 (2016) · Zbl 1329.05148
[29] Ostrom, T. G.; Wagner, A., On projective and affine planes with transitive collineation groups., Math. Z, 71, 186-199 (1959) · Zbl 0085.14302
[30] Payne, Stanley E.; Thas, Joseph A., Finite generalized quadrangles, EMS Series of Lectures in Mathematics, xii+287 pp. (2009), European Mathematical Society (EMS), Z\"{u}rich · Zbl 1247.05047
[31] Praeger, Cheryl E., Finite quasiprimitive graphs. Surveys in combinatorics, 1997 (London), London Math. Soc. Lecture Note Ser. 241, 65-85 (1997), Cambridge Univ. Press, Cambridge · Zbl 0881.05055
[32] Praeger, Cheryl E., An O’Nan-Scott theorem for finite quasiprimitive permutation groups and an application to \(2\)-arc transitive graphs, J. London Math. Soc. (2), 47, 2, 227-239 (1993) · Zbl 0738.05046
[33] Song S. J. Song, On the stabilisers of locally 2-transitive graphs, 1603.08398
[34] Tits, Jacques, Sur la trialit\'{e} et certains groupes qui s’en d\'{e}duisent, Inst. Hautes \'{E}tudes Sci. Publ. Math., 2, 13-60 (1959) · Zbl 0088.37204
[35] Toborg, Imke; Waldecker, Rebecca, Finite simple \(3^\prime \)-groups are cyclic or Suzuki groups, Arch. Math. (Basel), 102, 4, 301-312 (2014) · Zbl 1304.20022
[36] Vasil\cprime ev, A. V., Minimal permutation representations of finite simple exceptional groups of types \(G_2\) and \(F_4\), Algebra i Logika. Algebra and Logic, 35 35, 6, 371-383 (1996)
[37] Walter, John H., The characterization of finite groups with abelian Sylow \(2\)-subgroups, Ann. of Math. (2), 89, 405-514 (1969) · Zbl 0184.04605
[38] Wilson, Robert A., The finite simple groups, Graduate Texts in Mathematics 251, xvi+298 pp. (2009), Springer-Verlag London, Ltd., London · Zbl 1203.20012
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