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Flag-transitive $2$-$(v,k,λ)$ designs with $λ\ge (r,λ)^2$

Junchi Zhang, Jianbing Lu, Meizi Ou

Abstract

This paper is devoted to the study of $2$-designs with $λ\ge (r,λ)^2$ admitting a flag-transitive automorphism group $G$. The group $G$ has been shown to be point-primitive of either almost simple or affine type. In this paper, we classify the $2$-designs with $λ\geq (r,λ)^2>1$ admitting a flag-transitive almost simple automorphism group with socle $\mathrm{PSL}_n(q)$ or $\mathrm{PSU}_n(q)$ for $n \geq 3$.

Flag-transitive $2$-$(v,k,λ)$ designs with $λ\ge (r,λ)^2$

Abstract

This paper is devoted to the study of -designs with admitting a flag-transitive automorphism group . The group has been shown to be point-primitive of either almost simple or affine type. In this paper, we classify the -designs with admitting a flag-transitive almost simple automorphism group with socle or for .

Paper Structure

This paper contains 5 sections, 22 theorems, 59 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['main']} for $\mathrm{Soc}(G)=\mathrm{PSL}_n(q)$
  4. Proof of Theorem \ref{['main']} for $\mathrm{soc}(G)=\mathrm{PSU}_n(q)$
  5. Acknowledgements

Key Result

Theorem 1.1

Let $\mathcal{D}$ be a non-trivial $2$-$(v,k,\lambda)$ design with $\lambda\geq (r,\lambda)^2>1$, admitting a flag-transitive automorphism group $G$ with socle $\mathrm{PSL}_n(q)$ or $\mathrm{PSU}_n(q)$, where $n\geq3$ and $q$ is a prime power. Let $H=G_\alpha$ be the stabilizer of a point $\alpha$ If $\mathrm{Soc}(G)=\mathrm{PSU}_n(q)$, then $n=3$ and $H$ is the stabilizer of a totally singular

Theorems & Definitions (37)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • ...and 27 more