Block-Transitive Automorphism Groups of $2$-$(v,5,λ)$ Designs

TL;DR

This work resolves the structure of block-transitive automorphism groups acting on 2-$(v,5,λ)$ designs by splitting into point-imprimitive and point-primitive cases. It leverages imprimitivity partitions and base-block arguments to enumerate all admissible configurations when the group is imprimitive, proving four possible $(v,c,d)$ triples and delivering complete classifications (including explicit λ-values and unique designs) for $v=16$ and $v=21$, with partial results for $v=81$ due to computational limits. For the primitive case, the authors apply the O'Nan–Scott theorem to constrain the possible primitive types to affine, almost simple, or product type, and they show the nonexistence of twisted wreath and diagonal types; in the product-type realm they identify three socle configurations that yield explicit 2-$(v,5,λ)$ designs with $v=81$ or $v=361$ and corresponding λ-values. The combination of combinatorial design theory and finite permutation group theory yields a near-complete catalog of block-transitive 2-designs with small block size, advancing understanding of symmetry in incidence structures and providing concrete data for further study and applications.

Abstract

This paper investigates $2$-$(v,5,λ)$ designs $\mathcal{D}$ admitting a block-transitive automorphism group $G$. We first prove that if $G$ is point-imprimitive, then $v$ must be one of 16, 21, or 81. We further provide a complete classification of all such designs for $v=16$ and $v=21$. Secondly, we demonstrate that if $G$ is point-primitive, then it must be of affine type, almost simple type, or product type. Additionally, we present a classification of pairs $(\mathcal{D},G)$ where $G$ is of product type.

Theorems & Definitions (18)

• Definition 1.1
• Theorem 1
• Remark 1
• Theorem 2
• Theorem 3
• Lemma 2.1
• Corollary 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4