A New Near Octagon and the Suzuki Tower

Abstract

We construct and study a new near octagon of order (2, 10) which has its full automorphism group isomorphic to the group G 2 (4):2 and which contains 416 copies of the Hall-Janko near octagon as full subgeometries. Using this near octagon and its substructures we give geometric constructions of the G 2 (4)-graph and the Suzuki graph, both of which are strongly regular graphs contained in the Suzuki tower. As a subgeometry of this octagon we have discovered another new near octagon, whose order is (2, 4).

Characterizations of the Suzuki tower near polygons

“…In this section we will show that N is isomorphic to V A,B . Since HJ is a near octagon of order (2,4) with H(2) D isometrically embedded in it, this will prove Theorem 1.2.…”

“…The Suzuki tower (a name coined by Tits, as it has been mentioned in [19]) is the sequence of five finite simple groups L 3 (2) < U 3 (3) < J 2 < G 2 (4) < Suz where each group (except the last one) is a maximal subgroup of the next group in the sequence. The five groups in the Suzuki tower correspond to five vertex-transitive graphs Γ 0 , Γ 1 , Γ 2 , Γ 3 , Γ 4 where the automorphism groups of the Γ i 's are L 3 (2):2, U 3 (3):2, J 2 :2, G 2 (4):2 and Suz:2, respectively [26]. Using the properties of G that we have described in [4], we gave a correspondence between the Suzuki tower and the tower of four near polygons H(2, 1) ⊂ H(2) D ⊂ HJ ⊂ G, where H(2, 1) is the unique generalized hexagon of order (2, 1) (which is the point-line dual of the incidence graph of the Fano plane).…”

“…The five groups in the Suzuki tower correspond to five vertex-transitive graphs Γ 0 , Γ 1 , Γ 2 , Γ 3 , Γ 4 where the automorphism groups of the Γ i 's are L 3 (2):2, U 3 (3):2, J 2 :2, G 2 (4):2 and Suz:2, respectively [26]. Using the properties of G that we have described in [4], we gave a correspondence between the Suzuki tower and the tower of four near polygons H(2, 1) ⊂ H(2) D ⊂ HJ ⊂ G, where H(2, 1) is the unique generalized hexagon of order (2, 1) (which is the point-line dual of the incidence graph of the Fano plane). The first four simple groups in the Suzuki tower act as automorphism groups of the corresponding near polygons.…”

“…Theorem 1.2. The Hall-Janko near octagon is the unique near octagon of order (2,4) that contains the dual split Cayley hexagon of order (2,2) as an isometrically embedded subgeometry. Theorem 1.3.…”

“…In [3] we used the so-called valuation geometry of H(2) D , obtained with the help of a computer, to prove that it is not contained in any semi-finite near hexagon as a full isometrically embedded subgeometry, thus giving a partial answer to the famous open problem about existence of semi-finite generalized polygons in this particular case. Also, the G 2 (4) near octagon G was first constructed using the valuation geometry of HJ (see [4,Appendix]). The algorithm provided in [3] to compute the valuation geometry of H(2) D can also be used to compute the valuation geometry of HJ and both these valuation geometries will be crucial to our proofs.…”