# Finite exceptional groups of Lie type and symmetric designs

## TL;DR

This paper investigates symmetric designs with automorphism groups derived from finite simple exceptional Lie type groups, establishing parameter restrictions and identifying specific parameter sets for certain cases.

## Contribution

It provides a reduction theorem limiting possible parameters and shows that certain design parameters cannot be coprime or prime, with explicit examples for specific groups.

## Key findings

- Parameters $k$ and $\lambda$ are not coprime.
- Neither $k$ nor $\lambda$ can be prime.
- Explicit parameter sets identified for $G=G_2(3)$.

## Abstract

In this article, we study symmetric $(v, k, \lambda)$ designs admitting a flag-transitive and point-primitive automorphism group $G$ whose socle $X$ is a finite simple exceptional group of Lie type. We prove a reduction theorem, severely restricting the possible parameters of such designs. We also prove that the parameters $k$ and $\lambda$ are not coprime, and neither of these parameters can be prime. Moreover, if $\lambda$ is at most $100$, we show that there are two such parameters sets, namely, $(351,126,45)$ and $(378,117,36)$ for $G=X=G_{2}(3)$. Our analysis depends heavily on detailed information about actions of finite exceptional almost simple groups of Lie type on the cosets of their large maximal subgroups. In particular, properties derived in the paper about large subgroups and the subdegrees of such actions may be of independent interest.