# Flag-transitive block designs with prime replication number and almost simple groups

## TL;DR

This paper classifies 2-designs with prime replication numbers that admit flag-transitive automorphism groups, focusing on those with almost simple automorphism groups and point-primitive actions.

## Contribution

It provides a complete classification of such 2-designs with prime replication numbers under almost simple automorphism groups.

## Key findings

- Identifies all 2-designs with prime replication number and flag-transitive automorphism groups.
- Characterizes automorphism groups as point-primitive of almost simple or affine type.
- Determines all 2-designs with prime replication number admitting an almost simple automorphism group.

## Abstract

In this article, we study $2$-designs with prime replication number admitting a flag-transitive automorphism group. The automorphism groups of these designs are point-primitive of almost simple or affine type. We determine $2$-designs with prime replication number admitting an almost simple automorphism group.