Flag-Transitive Point-Primitive Symmetric $(v,k,\lambda)$ Designs with Large $k$

In 2012, Tian and Zhou conjectured that a flag-transitive and point-primitive automorphism group of a symmetric $(v,k,\lambda)$ design must be an affine or almost simple group. In this paper, we study this conjecture and prove that if $k\leqslant 10^3$ and $G\leqslant Aut(\mathcal{D})$ is flag-transitive, point-primitive, then $G$ is affine or almost simple. This support the conjecture.

Mean Proximality, Mean Sensitivity and Mean Li–Yorke Chaos for Amenable Group Actions

We consider mean proximality and mean Li–Yorke chaos for [Formula: see text]-systems, where [Formula: see text] is a countable discrete infinite amenable group. We prove that if a countable discrete infinite abelian group action is mean sensitive and there is a mean proximal pair consisting of a transitive point and a periodic point, then it is mean Li–Yorke chaotic. Moreover, we give some characterizations of mean proximal systems for general countable discrete infinite amenable groups.

Line-transitive point-imprimitive linear spaces with number of points being a product of two primes

The work studies the line-transitive point-imprimitive automorphism groups of finite linear spaces, and is underway on the situation when the numbers of points are products of two primes. Let [Formula: see text] be a non-trivial finite linear space with [Formula: see text] points, where [Formula: see text] and [Formula: see text] are two primes. We prove that if [Formula: see text] is line-transitive point-imprimitive, then [Formula: see text] is solvable.