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Flag-transitive point-primitive quasi-symmetric $2$-designs with block intersection numbers $0$ and $y\leq10$

Jianbing Lu, Yu Zhuang

Abstract

In this paper, we show that for a non-trivial quasi-symmetric $2$-design $\mathcal{D}$ with two block intersection numbers $x=0$ and $2\leq y\leq10$, if $G\leq \mathrm{Aut}(\mathcal{D})$ is flag-transitive and point-primitive, then $G$ is either of affine type or almost simple type. Moreover, we prove that the socle of $G$ cannot be an alternating group. If the socle of $G$ is a sporadic group, then $\mathcal{D}$ and $G$ must be one of the following: $\mathcal{D}$ is a $2$-$(12,6,5)$ design with block intersection numbers $0,3$ and $G=\mathrm{M}_{11}$, or $\mathcal{D}$ is a $2$-$(22,6,5)$ design with block intersection numbers $0,2$ and $G=\mathrm{M}_{22}$ or $\mathrm{M}_{22}:2$.

Flag-transitive point-primitive quasi-symmetric $2$-designs with block intersection numbers $0$ and $y\leq10$

Abstract

In this paper, we show that for a non-trivial quasi-symmetric -design with two block intersection numbers and , if is flag-transitive and point-primitive, then is either of affine type or almost simple type. Moreover, we prove that the socle of cannot be an alternating group. If the socle of is a sporadic group, then and must be one of the following: is a - design with block intersection numbers and , or is a - design with block intersection numbers and or .

Paper Structure

This paper contains 16 sections, 17 theorems, 31 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Flag-transitive $2$-designs
  4. Maximal subgroups of the almost simple groups with alternating socle
  5. Proof of Theorem \ref{['main']}
  6. Computational method
  7. Twisted wreath type
  8. Simple diagonal type
  9. Product action type
  10. Proof of Theorem \ref{['second']}
  11. $H$ acts primitively on $\Omega_{n}$
  12. $H$ acts transitively and imprimitively on $\Omega_{n}$
  13. $H$ acts intransitively on $\Omega_{n}$
  14. Proof of Theorem \ref{['third']}
  15. Declaration of competing interest
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $\mathcal{D}$ be a non-trivial quasi-symmetric $2$-design with block intersection numbers $x=0$ and $y\leq10$. If $G\leq\mathrm{Aut}(\mathcal{D})$ is flag-transitive and point-primitive, then $G$ must be either of affine type or almost simple type.

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • ...and 15 more