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On generalized hexagons and a near octagon whose lines have three points. (English) Zbl 0565.05014
The authors prove that the only generalized hexagons of order (2,2) are the classical one and its dual (both associated with the group $G\sb 2(2))$. They also prove the uniqueness of the generalized hexagon of order (2,8), associated with ${}\sp 3D\sb 4(2)$, and of the near octagon of order (2,4;0,3), associated with the Hall-Janko group. The treatment of the case (2,8) is facilitated by the use of results of Ronan and Timmesfeld. The arguments for all three cases are presented in terms of the (distance-regular) incidence graphs associated with the geometries. A central theme is the identification of subgraphs which are isomorphic to ”2-covers” of the n-cube.
Reviewer: D.A.Drake

05B30Other designs, configurations
20D08Simple groups: sporadic finite groups
51M20Polyhedra and polytopes; regular figures, division of spaces
05B25Finite geometries (combinatorics)
51D20Combinatorial geometries
Full Text: DOI
[1] Biggs, N. L.: Algebraic graph theory. (1974) · Zbl 0284.05101
[2] Cohen, A. M.: Geometries originating from certain distance-regular graphs. Proc. of finite geometries and designs, London math. Soc. lecture notes, 81-87 (1981) · Zbl 0472.05030
[3] Haemers, W.; Roos, C.: An inequality for generalized hexagons. Geometriae dedicata 10, 219-222 (1981) · Zbl 0463.51012
[4] Ronan, M. A.: A note on the $3 D4(q)$ generalized hexagons. J. comb. Theory, ser A 29, 249-250 (1980) · Zbl 0444.05028
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[8] Tits, J.: Buildings of spherical type and finite BN pairs. Springer lecture notes 386 (1974) · Zbl 0295.20047