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On flag-transitive automorphism groups of $2$-designs with $λ$ prime

Seyed Hassan Alavi, Ashraf Daneshkhah, Alessandro Montinaro

TL;DR

The paper classifies flag-transitive and point-primitive 2-designs with prime λ whose almost-simple automorphism groups have socle either a finite exceptional simple group or a sporadic simple group. For exceptional socles, it yields two infinite families, including the Suzuki–Tits ovoid design, and a second family with explicit parameter sets; for the G$_2(q)$ case with even q and SU$_3(q)$-type stabilizers, it constructs a unique design realized as the coset geometry $ extsf{Cos}(T,H,K)$ with parameter set $(v,b,r,k,λ)=(q^{3}(q^{3}-1)/2,(q+1)(q^{6}-1),(q+1)(q^{3}+1),q^{3}/2,q+1)$, where $q+1$ is a Fermat prime. In the sporadic realm, computational analysis yields exactly three designs with parameter sets $(176,1100,50,8,2)$, $(12,22,11,6,5)$, and $(22,77,21,6,5)$. These results advance understanding of symmetry in combinatorial designs and connect to coset-geometry constructions and Suzuki–Tits configurations.

Abstract

In this article, we study $2$-$(v,k,λ)$ designs $\mathcal{D}$ with $λ$ prime admitting flag-transitive and point-primitive almost simple automorphism groups $G$ with socle $T$ a finite exceptional simple group or a sporadic simple groups. If the socle of $G$ is a finite exceptional simple group, then we prove that $\mathcal{D}$ is isomorphic to one of two infinite families of $2$-designs with point-primitive automorphism groups, one is the Suzuki-Tits ovoid design with parameter set $(v,b,r,k,λ)=(q^{2}+1,q^{2}(q^{2}+1)/(q-1),q^{2},q,q-1)$ design, where $q-1$ is a Mersenne prime, and the other is newly constructed in this paper and has parameter set $(v,b,r,k,λ)=(q^{3}(q^{3}-1)/2,(q+1)(q^{6}-1),(q+1)(q^{3}+1),q^{3}/2,q+1)$, where $q+1$ a Fermat prime. If $T$ is a sporadic simple group, then we show that $\mathcal{D}$ is isomorphic to a unique design admitting a point-primitive automorphism group with parameter set $(v,b,r,k,λ)=(176,1100,50,2)$, $(12,22,11,6,5)$ or $(22,77,21,6,5)$.

On flag-transitive automorphism groups of $2$-designs with $λ$ prime

TL;DR

The paper classifies flag-transitive and point-primitive 2-designs with prime λ whose almost-simple automorphism groups have socle either a finite exceptional simple group or a sporadic simple group. For exceptional socles, it yields two infinite families, including the Suzuki–Tits ovoid design, and a second family with explicit parameter sets; for the G case with even q and SU-type stabilizers, it constructs a unique design realized as the coset geometry with parameter set , where is a Fermat prime. In the sporadic realm, computational analysis yields exactly three designs with parameter sets , , and . These results advance understanding of symmetry in combinatorial designs and connect to coset-geometry constructions and Suzuki–Tits configurations.

Abstract

In this article, we study - designs with prime admitting flag-transitive and point-primitive almost simple automorphism groups with socle a finite exceptional simple group or a sporadic simple groups. If the socle of is a finite exceptional simple group, then we prove that is isomorphic to one of two infinite families of -designs with point-primitive automorphism groups, one is the Suzuki-Tits ovoid design with parameter set design, where is a Mersenne prime, and the other is newly constructed in this paper and has parameter set , where a Fermat prime. If is a sporadic simple group, then we show that is isomorphic to a unique design admitting a point-primitive automorphism group with parameter set , or .
Paper Structure (4 sections, 6 theorems, 10 equations, 3 tables)

This paper contains 4 sections, 6 theorems, 10 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Definitions and notation
  3. Proof of Theorem \ref{['thm:exp']}
  4. Sporadic simple groups

Key Result

Theorem 1.1

Let $\mathcal{D}$ be a nontrivial $2$-$(v,k,\lambda)$ design with $\lambda$ prime admitting a flag-transitive and point-primitive automorphism group $G$ with socle $T$ a finite exceptional simple group. Then (up to isomorphism) one of the following holds

Theorems & Definitions (12)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 2 more