Suppose that an automorphism group G acts flag-transitively on a finite generalized hexagon or octagon S, and suppose that the action on both the point and line set is primitive. We show that G is an almost simple group of Lie type, that is, the socle of G is a simple Chevalley group.

A class of translation affine planes of order q2, where q is a power of a prime/>>3 is constructed. These planes have an interesting property that their collineation groups are flag transitive.

In the paper we show that all combinatorial triangle-free configurations for v ≤ 18 are geometrically realizable. We also show that there is a unique smallest astral (183) triangle-free configuration and its Levi graph is the generalized Petersen graph G(18, 5). In addition, we present geometric realizations of the unique flag transitive triangle-free configuration (203) and the unique point tr...

A nite C 3-geometry is called anomalous if it is neither a building nor the A 7-geometry. It is conjectured that no ag-transitive thick anomalous C 3-geometry exists. For a ag-transitive thick anomalous C 3-geometry, we prove that its 2-order y is odd and that its full automorphism group is non-solvable. As a corollary, there are no ag-transitive circular extensions of duals of anomalous C 3-ge...

We study flat flag-transitive c.c∗-geometries. We prove that, apart from one exception related to Sym(6), all these geometries are gluings in the meaning of [6]. They are obtained by gluing two copies of an affine space over GF(2). There are several ways of gluing two copies of the n-dimensional affine space over GF(2). In one way, which deserves to be called the canonical one, we get a geometr...

Two families of flag-transitive nondesarguesian affine planes of odd order are defined, and isomorphisms among the various planes are studied. Thir ty years ago Wagner [5] proved the impor tan t result that every finite flag-transitive affine plane is a t ranslat ion plane. However , since that t ime relatively few classes of such planes have been found. The purpose of this note is to c o m m e...

(with 1 , q , ̀ ) . An infinite family of simply connected examples for this diagram with q 5 2 is obtained by truncating Coxeter complexes of type D n . Two dif ferent finite families with three members and q 5 4 arise from the Steiner systems for the Mathieu groups M 2 2 , M 2 3 and M 2 4 . The geometries of these two families are simply connected . Three more simply connected examples of rank...

are considered in this paper. Examples of finite Qn-geometries are constructed for any n i> 2. In particular, we obtain new examples of finite 2-arc-transitive graphs of girth 5, containing Petersen subgraphs (cf. [4], [5] for an enquiry into this class of graphs). Furthermore, we give a complete classification of Q2-geometries using coset enumeration. In particular, we show that they are all f...

A pseudo-hyperoval of a projective space PG(3n− 1, q), q even, is a set of q +2 subspaces of dimension n − 1 such that any three span the whole space. We prove that a pseudo-hyperoval with an irreducible transitive stabiliser is elementary. We then deduce from this result a classification of the thick generalised quadrangles Q that admit a point-primitive, line-transitive automorphism group wit...