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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Notes
See http://brauer.maths.qmul.ac.uk/Atlas/v3/exc/G24/ for the notation and description of this class.
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This is one of several papers published in Designs, Codes and Cryptography comprising the special issue in honor of Andries Brouwer’s 65th birthday.
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Bishnoi, A., Bruyn, B.D. Characterizations of the Suzuki tower near polygons. Des. Codes Cryptogr. 84, 115–133 (2017). https://doi.org/10.1007/s10623-016-0230-7
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DOI: https://doi.org/10.1007/s10623-016-0230-7