The rank 3 residually connected flag transitive geometries Γ for M22 for which the stabilizer of each object in Γ is a maximal subgroup of M22 are determined. As a result this deals with the infelicities in Theorem 3 of Kilic and Rowley, On rank 2 and rank 3 residually connected geometries for M22. Note di Matematica, 22(2003), 107–154.

It has been shown that if a (v,k,λ )-symmetric design with λ ≤ 3 admits a flag-transitive automorphism group G which acts primitively on points, then G must be of affine or almost simple type. Here we extend the result to λ = 4.

We show that ifG is a group of automorphisms of a thick finite generalised quadrangleQ acting primitively on both the points and lines of Q, then G is almost simple. Moreover, if G is also flag-transitive then G is of Lie type.

Some old and new constructions of the tilde geometry (the flag transitive connected triple cover of the unique generalized quadrangle W(2) of order (2, 2)) are discussed. Using them, we prove some properties of that geometry. In particular, we compute its generating rank, we give an explicit description of its universal projective embedding and we determine its homogeneous embeddings.

The main aim of this article is to study (v, k, λ)-symmetric designs admitting a flag-transitive and point-primitive automorphism group G whose socle is PSL(3, q). We indeed show that the only possible design satisfying these conditions is a Desarguesian projective plane PG(2, q) and G ⩾ PSL(3, q).

In this paper we develop general techniques for enumerating orthogonal Buekenhout unitals embedded in two-dimensional translation planes. We then apply these techniques in the regular nearfield planes, the odd-order Hall planes, and the odd-order flag-transitive affine planes. Stabilizers of the resulting unitals also are computed.

We classify flips of buildings arising from non-degenerate unitary spaces of dimension at least 4 over finite fields of odd characteristic in terms of their action on the underlying vector space. We also construct certain geometries related to flips and prove that these geometries are flag transitive. Corresponding author: [email protected] 1