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Homogeneous designs and geometric lattices. (English) Zbl 0559.05015

Theorem 1. Let \({\mathcal D}\) be a design with \(\lambda =1\) admitting an automorphism group 2-transitive on points. Then \({\mathcal D}\) is one of the following designs: (i) PG(d,q), (ii) AG(d,q), (iii) The design with \(v=q^ 3+1\) and \(k=q+1\) associated with PSU(3,q) or \({}^ 2G_ 2(q)\), (iv) One of two affine planes, having \(3^ 4\) or \(3^ 6\) points [P. Dembowski, Finite geometries (1968; Zbl 0159.500)] (v), One of two designs having \(v=3^ 6\) and \(k=3^ 2\) (C. Hering, to appear). Theorem 2. Let \({\mathcal L}\) be a finite geometric lattice of rank at least 3 such that Aut \({\mathcal L}\) is transitive on ordered bases. Then either (i) \({\mathcal L}\) is a truncation of a Boolean lattice or a projective or affine geometry, (ii) \({\mathcal L}\) is the lattice associated with a Steiner system S(3,6,22), S(4,7,23), or S(5,8,24), or (iii) \({\mathcal L}\) is the lattice associated with the 65-point design for PSU(3,4). The groups in these theorems are described in the course of the proofs, which are a consequence of the classification of all finite simple groups [D. Gorenstein, Finite simple groups: An introduction to their classification (1982; Zbl 0483.20008)].
Reviewer: W.Moser

MSC:

05B25 Combinatorial aspects of finite geometries
51D20 Combinatorial geometries and geometric closure systems
05B07 Triple systems
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