The Erd\H{o}s-Ko-Rado Theorem for non-quasiprimitive groups of degree $3p$

TL;DR

This paper investigates the intersection density of non-quasiprimitive transitive groups of degree 3p, confirming a conjecture for most cases and identifying specific exceptions related to Fermat primes and almost simple groups.

Contribution

It extends the understanding of intersection density for transitive groups of degree 3p, proving the conjecture for non-quasiprimitive groups with certain specific conditions.

Findings

Confirmed the intersection density conjecture for non-quasiprimitive groups of degree 3p.

Identified potential exceptions involving Fermat primes and almost simple groups.

Extended previous results from primitive to non-quasiprimitive groups.

Abstract

The \emph{intersection density} of a finite transitive group $G\leq \operatorname{Sym}(\Omega)$ is the rational number $\rho(G)$ given by the ratio between the maximum size of a subset of $G$ in which any two permutations agree on some elements of $\Omega$ and the order of a point stabilizer of $G$. In 2022, Meagher asked whether $\rho(G)\in \{1,\frac{3}{2},3\}$ for any transitive group $G$ of degree $3p$, where $p\geq 5$ is an odd prime. For the primitive case, it was proved in [\emph{J. Combin. Ser. A}, 194:105707, 2023] that the intersection density is $1$. It is shown in this paper that the answer to this question is affirmative for non-quasiprimitive groups, unless possibly when $p = q+1$ is a Fermat prime and $\Omega$ admits a unique $G$-invariant partition $\mathcal{B}$ such that the induced action $\overline{G}_\mathcal{B}$ of $G$ on $\mathcal{B}$ is an almost simple group containing $\operatorname{PSL}_{2}(q)$.