# On flag-transitive automorphism groups of symmetric designs

## TL;DR

This paper classifies flag-transitive automorphism groups of symmetric designs with specific divisibility and size conditions, showing they are either point-primitive of certain types or point-imprimitive with explicit parameters.

## Contribution

It provides a complete classification of such automorphism groups under the given conditions, including explicit parameter cases and examples.

## Key findings

- Automorphism groups are either point-primitive of affine or almost simple type or point-imprimitive.
- Explicit parameters for the point-imprimitive case are given as functions of λ.
- Examples are provided for both classification cases.

## Abstract

In this article, we study flag-transitive automorphism groups of non-trivial symmetric $(v, k, \lambda)$ designs, where $\lambda$ divides $k$ and $k\geq \lambda^2$. We show that such an automorphism group is either point-primitive of affine or almost simple type, or point-imprimitive with parameters $v=\lambda^{2}(\lambda+2)$ and $k=\lambda(\lambda+1)$, for some positive integer $\lambda$. We also provide some examples in both possibilities.