On the Largest intersecting set in $GL_2(q)$ and some of its subgroups

TL;DR

This paper establishes an Erdős-Ko-Rado type theorem for intersecting sets within certain subgroups of the general linear group over a finite field, characterizing maximum intersecting sets explicitly.

Contribution

It proves a new Erdős-Ko-Rado theorem for subgroups of GL_2(q) and classifies all maximum intersecting sets in these groups.

Findings

Maximum intersecting sets are either cosets of point stabilizers or specific subgroups.

Every intersecting set is contained within a maximum intersecting set.

Provides explicit descriptions of maximum intersecting sets in G.

Abstract

Let $q$ be a power of a prime number and $V$ be the $2$-dimensional column vector space over a finite field $\mathbb{F}_{q}$. Assume that $SL_2(V)<G\leq GL_2(V)$. In this paper we prove an Erd{\H{o}}s-Ko-Rado theorem for intersecting sets of G and we show that every maximum intersecting set of $G$ is either a coset of the stabilizer of a point or a coset of $\mathcal{G}_{\langle w\rangle}$, where $\mathcal{G}_{\langle w\rangle}=\{M\in G:\forall v\in V, Mv-v\in \langle w\rangle\}$, for some $w\in V\setminus \{0\}$. It is also shown that every intersecting set of $G$ is contained in a maximum intersecting set.