Erd\H{o}s-Ko-Rado results for the general linear group, the special linear group and the affine general linear group

TL;DR

This paper proves Erdős–Ko–Rado properties for the general linear, special linear, and affine general linear groups using algebraic methods, including adjacency matrices and Hoffman's ratio bound.

Contribution

It establishes EKR and EKR-module properties for these groups and introduces algebraic techniques to analyze intersecting sets in these algebraic structures.

Findings

Both $ ext{GL}(q)$ and $ ext{SL}(q)$ have the EKR property.

The groups also possess the EKR-module property.

Analysis of 2-intersecting sets in $ ext{PGL}(2,q)$.

Abstract

In this paper, we show that both the general linear group $\gl{q}$ and the special linear group $\slg{q}$ have both the EKR property and the EKR-module property. This is done using an algebraic method; a weighted adjacency matrix for the derangement graph for the group is found and Hoffman's ratio bound is applied to this matrix. We also consider the group $\agl{q}$ and the 2-intersecting sets in $\PGL(2,q)$.