Block-transitive t-designs, II: large t

Summary

Abstract

We study block-transitive t-(v, k, λ) designs for large t. We show that there are no nontrivial block-transitive 8-designs, and no nontrivial flag-transitive 7-designs. There are no known nontrivial block-transitive 6-designs; we show that the automorphism group of such a design, or of a flag-transitive 5-design with more than 24 points, must be either an affine group over GF(2) or a 2-dimensional projective linear group. We begin the investigation of these two cases, and construct a flag-transitive 5-(256, 24, λ) design for a suitable value of λ.

Introduction

A t-(v, k, λ) design is a pair D = (X, B), where X is a set of v points, B a set of k-element subsets of X called blocks, such that any t points are contained in exactly λ blocks, for some t ≤ k and λ > 0. Such a design (X, B) is said to be trivial if B consists of all the k-element subsets of X. A flag in a design D is an incident point-block pair. A subgroup G of the automorphism group of D is said to be block-transitive if G is transitive on B ; D is block-transitive if Aut(D) is. Point- and flag-transitivity are defined similarly. For information about t-designs, see Hughes and Piper [11].

In this paper we consider nontrivial block-transitive t-designs with t large. We use a result of Ray-Chaudhuri and Wilson [16] together with the finite simple group classification to show in Section 2 that t ≤ 7.