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

TL;DR

This paper classifies 2-(v,5,λ) designs with block-transitive automorphism groups, detailing the possible parameters and types of automorphism groups, and providing complete classifications for certain cases.

Contribution

It provides a complete classification of 2-(v,5,λ) designs with block-transitive automorphism groups for specific parameters and types, advancing understanding of their symmetry properties.

Findings

If G is point-imprimitive, then v is 16, 21, or 81.

Complete classification of designs for v=16 and v=21.

If G is point-primitive, it must be of affine, almost simple, or product type.

Abstract

This paper investigates $2$-$(v,5,\lambda)$ 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.