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

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 $\operatorname{PSL}_{2}(q)$.