An Erd\H{o}s-Ko-Rado theorem in general linear groups

Let $S_n$ be the symmetric group on $n$ points. Deza and Frankl [M. Deza and P. Frankl, On the maximum number of permutations with given maximal or minimal distance, J. Combin. Theory Ser. A 22 (1977) 352–360] proved that if ${\cal F}$isanintersectingsetin$S_n$then$|{\cal F}|\leq(n-1)!$. In this paper we consider the $q$-analogue version of this result. Let ${\mathbb{F}}_q^n$ be the $n$-dimensional row vector space over a finite field ${\mathbb{F}}_q$ and $G{L}_n({\mathbb{F}}_q)$ the general linear group of degree $n$. A set ${\cal F}_q\subseteq GL_n($\mathbb{F}$_q)$isintersectingifforany$T,S\in{\cal F}_q$thereexistsanon-zerovector$\alpha\in $\mathbb{F}$_q^n$suchthat$\alpha T=\alpha S$\mathrm{.}Let${\cal F}_q$beanintersectingsetin$GL_n($\mathbb{F}$_q)$\mathrm{.}Weshowthat$|{\cal F}q|\leq q^{(n-1)n/2}\prod{i=1}^{n-1}(q^i-1)$.