Intersection density of imprimitive groups of degree $pq$

TL;DR

This paper investigates the intersection density of imprimitive groups of degree pq, revealing conditions under which the density exceeds 1, especially involving cyclic codes and group structures like almost simple groups.

Contribution

It provides new insights into the intersection density of imprimitive groups of degree pq, linking it to cyclic codes and group classifications such as almost simple groups.

Findings

Intersection density exceeds 1 when certain cyclic codes exist.

For quasiprimitive groups, cases reduce to almost simple groups containing specific subgroups.

Examples show some groups have intersection density exactly 1 under certain conditions.

Abstract

A subset $\mathcal{F}$ of a finite transitive group $G\leq \operatorname{Sym}(\Omega)$ is \emph{intersecting} if any two elements of $\mathcal{F}$ agree on an element of $\Omega$. The \emph{intersection density} of $G$ is the number $$\rho(G) = \max\left\{ \mathcal{|F|}/|G_\omega| \mid \mathcal{F}\subset G \mbox{ is intersecting} \right\},$$ where $\omega \in\Omega$ and $G_\omega$ is the stabilizer of $\omega$ in $G$. It is known that if $G\leq \operatorname{Sym}(\Omega)$ is an imprimitive group of degree a product of two odd primes $p>q$ admitting a block of size $p$ or two complete block systems, whose blocks are of size $q$, then $\rho(G) = 1$. In this paper, we analyse the intersection density of imprimitive groups of degree $pq$ with a unique block system with blocks of size $q$ based on the kernel of the induced action on blocks. For those whose kernels are non-trivial, it is proved that the intersection density is larger than $1$ whenever there exists a cyclic code $C$ with parameters $[p,k]_q$ such that any codeword of $C$ has weight at most $p-1$, and under some additional conditions on the cyclic code, it is a proper rational number. For those that are quasiprimitive, we reduce the cases to almost simple groups containing $\operatorname{Alt}(5)$ or a projective special linear group. We give some examples where the latter has intersection density equal to $1$, under some restrictions on $p$ and $q$.