# The structure of $3$-pyramidal groups

## TL;DR

This paper classifies groups acting regularly on the points of 3-pyramidal combinatorial block designs, especially Kirkman triple systems, focusing on their involution structure.

## Contribution

It provides a classification of groups with specific involution properties acting on 3-pyramidal designs, extending understanding of their automorphism groups.

## Key findings

- Groups with this property have exactly 3 involutions, all conjugate.
- Classification results for groups acting on Kirkman triple systems.
- Insights into the automorphism structure of 3-pyramidal designs.

## Abstract

A combinatorial block design $D$ is called $3$-pyramidal if there exists a subgroup $G$ of $\mbox{Aut}(D)$ fixing $3$ points and acting regularly on the other points. If this happens, we say that the design is $3$-pyramidal under $G$. In case $D$ is a Kirkman triple system, it is known that such a group $G$ has precisely $3$ involutions, all conjugate to each other. In this paper, we obtain a classification of the groups with this property.