# Classification of flag-transitive $2$-$(v, k, \lambda)$ designs with alternating group $A_n$($n \le 10$) as socle

## TL;DR

This paper classifies all flag-transitive, point-primitive 2-designs with alternating groups of degree up to 10 as socles, identifying 87 unique designs and advancing the understanding of design symmetry under almost simple groups.

## Contribution

It provides a complete classification of such designs with small alternating group socles, extending existing theory and offering a comprehensive reference for future research.

## Key findings

- 87 distinct designs classified up to isomorphism
- Enhanced understanding of automorphism groups of 2-designs
- Improved classification framework for designs with almost simple group actions

## Abstract

This paper is devoted to the classification of all flag-transitive point-primitive non-trivial $2$-$(v, k, \lambda)$ designs with the alternating group $A_n$($n \le 10$) as the socle of their automorphism groups, and 87 different designs are obtained up to isomorphism. The results of this study further improve the classification theory of designs under the action of almost simple groups, and provide reference for the follow-up study of similar problems.