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2-(v,k,3) designs admitting an almost simple, flag-transitive automorphism group with socle PSL(2,q)

Hongxue Liang, Zhihui Liu, Alessandro Montinaro

TL;DR

The paper addresses the classification of non-trivial $2$-$(v,k,3)$ designs with a flag-transitive automorphism group whose socle is $PSL(2,q)$. It develops a framework using primitive action results, maximal-subgroup data, and subdegree constraints to limit possibilities, supplemented by Magma computations to exclude remaining cases. The main result yields a five-case list, with two explicit realizations: the complete $2$-$(5,3,3)$ design and the $2$-$(26,6,3)$ Baer-subline design, while the other cases do not yield valid designs. This completes the lambda=3, flag-transitive PSL$(2,q)$–style classification and extends prior work on lambda=1,2 cases, connecting to known geometric configurations and Paley-type structures.

Abstract

In this paper, we completely classify the non-trivial 2-(v,k,3) designs admitting an almost simple, flag-transitive automorphism group with socle PSL(2,q).

2-(v,k,3) designs admitting an almost simple, flag-transitive automorphism group with socle PSL(2,q)

TL;DR

The paper addresses the classification of non-trivial - designs with a flag-transitive automorphism group whose socle is . It develops a framework using primitive action results, maximal-subgroup data, and subdegree constraints to limit possibilities, supplemented by Magma computations to exclude remaining cases. The main result yields a five-case list, with two explicit realizations: the complete - design and the - Baer-subline design, while the other cases do not yield valid designs. This completes the lambda=3, flag-transitive PSL–style classification and extends prior work on lambda=1,2 cases, connecting to known geometric configurations and Paley-type structures.

Abstract

In this paper, we completely classify the non-trivial 2-(v,k,3) designs admitting an almost simple, flag-transitive automorphism group with socle PSL(2,q).
Paper Structure (3 sections, 15 theorems, 29 equations, 2 tables)

This paper contains 3 sections, 15 theorems, 29 equations, 2 tables.

Table of Contents

  1. Introduction and Main Result
  2. Preliminaries
  3. Proof of the main result

Key Result

Theorem 1.1

Let $\mathcal{D}$ be a non-trivial $2$-design with $\lambda=3$ admitting $\mathrm{PSL}(2,q) \unlhd G \leqslant \mathrm{P \Gamma L(2,q)}$ as a flag-transitive automorphism group. Then one of the following holds:

Theorems & Definitions (30)

  • Theorem 1.1
  • Example 1.2
  • proof
  • Example 1.3
  • proof
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • ...and 20 more