On the Largest intersecting set in $G{L}_2(q)$ and some of its subgroups

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 $S{L}_2(V)<G\le G{L}_2(V)$. In this paper we prove an Erd{ő}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 ${\mathcal{G}}_{⟨w⟩}$, where ${\mathcal{G}}_{⟨w⟩}=\{M\in G:\forall v\in V,Mv-v\in ⟨w⟩\}$, for some $w\in V\setminus \{0\}$. It is also shown that every intersecting set of $G$ is contained in a maximum intersecting set.

arXiv commentThe title has been changed. Also, some parts of the paper were reorganized. Accepted for publication in Comptes Rendus Mathematique