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

TL;DR

This paper investigates the intersection density of primitive and imprimitive groups of degree pq, where p and q are odd primes, establishing that the maximum intersection density is 1 under certain conditions.

Contribution

It proves that for imprimitive groups of degree pq with multiple systems of imprimitivity and for primitive groups with certain socle properties, the intersection density equals 1.

Findings

For imprimitive groups with at least two systems of imprimitivity, intersection density is 1.

For primitive groups with socle admitting an imprimitive subgroup, intersection density is 1.

Provides conditions under which the intersection density reaches its maximum value of 1.

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.