On the intersection density of primitive groups of degree a product of two odd primes

Abstract

A subset $\mathcal{F}$ of a finite transitive group $G\leq \operatorname{Sym}(\Omega)$ is intersecting if for any $g,h\in \mathcal{F}$ there exists $\omega \in \Omega$ such that $\omega^g = \omega^h$. The \emph{intersection density} $\rho(G)$ of $G$ is the maximum of $\left\{ \frac{|\mathcal{F}|}{|G_\omega|} \mid \mathcal{F}\subset G \mbox{ is intersecting} \right\}$, where $G_\omega$ is the stabilizer of $\omega$ in $G$. In this paper, it is proved that if $G$ is an imprimitive group of degree $pq$, where $p$ and $q$ are distinct odd primes, with at least two systems of imprimitivity then $\rho(G) = 1$. Moreover, if $G$ is primitive of degree $pq$, where $p$ and $q$ are distinct odd primes, then it is proved that $\rho(G) = 1$, whenever the socle of $G$ admits an imprimitive subgroup.