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Designs, Codes and Cryptography
Published in: Designs, Codes and Cryptography 1-2/2017

01-06-2016

Characterizations of the Suzuki tower near polygons

Authors: Anurag Bishnoi, Bart De Bruyn

Published in: Designs, Codes and Cryptography | Issue 1-2/2017

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Abstract

In recent work, we constructed a new near octagon \(\mathcal {G}\) from certain involutions of the finite simple group \(G_2(4)\) and showed a correspondence between the Suzuki tower of finite simple groups, \(L_3(2)< U_3(3)< J_2< G_2(4) < Suz\), and the tower of near polygons, \(\mathrm {H}(2,1) \subset \mathrm {H}(2)^D \subset \mathsf {HJ} \subset \mathcal {G}\). Here we characterize each of these near polygons (except for the first one) as the unique near polygon of the given order and diameter containing an isometrically embedded copy of the previous near polygon of the tower. In particular, our characterization of the Hall–Janko near octagon \(\mathsf {HJ}\) is similar to an earlier characterization due to Cohen and Tits who proved that it is the unique regular near octagon with parameters (2, 4; 0, 3), but instead of regularity we assume existence of an isometrically embedded dual split Cayley hexagon, \(\mathrm {H}(2)^D\). We also give a complete classification of near hexagons of order (2, 2) and use it to prove the uniqueness result for \(\mathrm {H}(2)^D\).
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Footnotes
1
See http://​brauer.​maths.​qmul.​ac.​uk/​Atlas/​v3/​exc/​G24/​ for the notation and description of this class.
 
Literature
1.
Bardoe M.K.: The universal embedding for the involution geometry of the Suzuki sporadic simple group. J. Algebra 186, 447–460 (1996).
2.
Bardoe M.K.: The universal embedding for the involution geometry of \(Co_1\). J. Algebra 217, 555–572 (1999).
3.
4.
Bishnoi A., De Bruyn B.: A new near octagon and the Suzuki tower. Electron. J. Comb. 23, #P2.35 (2016).
5.
6.
7.
Brouwer A.E., Cohen A.M., Hall J.I., Wilbrink H.A.: Near polygons and Fischer spaces. Geom. Dedicata 49, 349–368 (1994).
8.
Brouwer A.E., Cohen A.M., Neumaier A.: Distance-Regular Graphs. Springer, Berlin (1989).
9.
Brouwer A.E., Fon-der-Flaass D.G., Shpectorov S.V.: Locally co-Heawood graphs. In: Finite Geometry and Combinatorics (Deinze, 1992). London Mathematical Society Lecture Note Series, vol. 191, pp. 59–68. Cambridge University Press, Cambridge (1993).
10.
Brouwer A.E., Wilbrink H.A.: The structure of near polygons with quads. Geom. Dedicata 14, 145–176 (1983).
11.
Cameron P.J.: Dual polar spaces. Geom. Dedicata 12, 75–85 (1982).
12.
Cohen A.M.: Geometries originating from certain distance-regular graphs. In: Finite Geometries and Designs (Proc. Conf., Chelwood Gate, 1980). London Mathematical Society Lecture Note Series, vol. 49, pp. 81–87. Cambridge University Press, Cambridge (1981).
13.
Cohen A.M., Tits J.: On generalized hexagons and a near octagon whose lines have three points. Eur. J. Comb. 6, 13–27 (1985).
14.
De Bruyn B.: Near Polygons. Frontiers in Mathematics. Birkhäuser, Basel (2006).
15.
De Bruyn B.: The valuations of the near polygon \({\cal G}_n\). Electron. J. Comb. 16 (2009) (Research Paper 137).
16.
De Bruyn B.: The use of valuations for classifying point-line geometries. In: Groups of Exceptional Type, Coxeter Groups and Related Geometries. Springer Proceedings in Mathematics & Statistics, vol. 82, pp. 27–40. Springer, Berlin (2014).
17.
De Bruyn B., Vandecasteele P.: Valuations of near polygons. Glasg. Math. J. 47, 347–361 (2005).
18.
19.
Leemans D.: A family of geometries related to the Suzuki tower. Commun. Algebra 33, 2201–2217 (2005).
20.
Neumaier A.: Rectagraphs, diagrams, and Suzuki’s sporadic simple group. In: Algebraic and Geometric Combinatorics. North-Holland Mathematics Studies, vol. 65, pp. 305–318. North-Holland, Amsterdam (1982).
21.
Pasechnik D.V.: Geometric characterization of graphs from the Suzuki chain. Eur. J. Comb. 14, 491–499 (1993).
22.
Payne S.E., Thas J.A.: Finite Generalized Quadrangles. European Mathematical Society, Zürich (2009).
23.
Shult E.E., Yanushka A.: Near \(n\)-gons and line systems. Geom. Dedicata 9, 1–72 (1980).
24.
Soicher L.H.: On simplicial complexes related to the Suzuki sequence graphs. In: Groups, Combinatorics and Geometry (Durham, 1990). London Mathematical Society Lecture Note Series, vol. 165, pp. 240–248. Cambridge University Press, Cambridge (1992).
25.
Soicher L.H.: Three new distance-regular graphs. Eur. J. Comb. 14, 501–505 (1993).
26.
Suzuki M.: A finite simple group of order 448, 345, 497, 600. In: Brauer H., Sag D. (eds.) Symposium on Finite Groups, pp. 113–119. Benjamin, New York (1969).
27.
Wilson, R.A.: The Finite Simple Groups. Graduate Texts in Mathematics, vol. 251. Springer, London (2009).
28.
Yoshiara S.: A lattice theoretical construction of a GAB of the Suzuki sporadic simple group. J. Algebra 112, 198–239 (1988).
Metadata
Title
Characterizations of the Suzuki tower near polygons
Authors
Anurag Bishnoi
Bart De Bruyn
Publication date
01-06-2016
Publisher
Springer US
Published in
Designs, Codes and Cryptography / Issue 1-2/2017
Print ISSN: 0925-1022
Electronic ISSN: 1573-7586
DOI
https://doi.org/10.1007/s10623-016-0230-7

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