Analysing flag-transitive point-imprimitive 2-designs

Abstract

In this paper we develop several general methods for analysing flag-transitive pointimprimitive 2-designs, which give restrictions on both the automorphisms and parameters of such designs. These constitute a tool-kit for analysing these designs and their groups. We apply these methods to complete the classification of flag-transitive, point-imprimitive 2-(v, k, λ) designs with λ at most 4.

Chain-imprimitive, flag-transitive 2-designs

“…So assume that 1 ≤ i < s. Since B is the disjoint union of all the intersections B ∩ C with C ∈ C i (B), we have C∈C i (B) y i = C∈C i (B) |B ∩ C| = |B| = k. Also, for any C ∈ C i , we have x C = y i if C ∈ C i (B), and x C = 0 otherwise; and since D is a 2-design, it follows from Theorem 2.2 that conditions Eqs. (10) and (11) in that theorem hold. We will show by induction that y i = y i for each i ∈ {1, .…”