The intersection density of non-quasiprimitive groups of degree 3p

Abstract

The intersection density of a finite transitive group $G\le \mathrm{Sym}\left(\Omega \right)$ is the rational number $\rho \left(G\right)$ given by the ratio between the maximum size of a subset of G in which any two permutations agree on some elements of Ω and the order of a point stabilizer of G. In 2022, Meagher asked whether $\rho \left(G\right)\in \{1,\frac{3}{2},3\}$ for any transitive group G of degree 3p, where $p\ge 5$ is an odd prime. If $G\le \mathrm{Sym}\left(\Omega \right)$ is transitive such that $\left|\Omega \right|=3p$, then it is known that $\rho \left(G\right)=1$ whenever (a) G is primitive or (b) G is imprimitive and admits a block of size p or at least two G-invariant partitions of Ω. In order to answer Meagher's question, it is left to analyze the intersection density of groups G admitting a unique G-invariant partition $B$ whose blocks are of size 3. If G is such a group and $\bar{G}$ is the group induced by the action of G on $B$, then we denote the kernel of the canonical epimorphism $G\to \bar{G}$ by $\ker (G\to \bar{G})$. The subgroup $\ker (G\to \bar{G})$ is trivial if and only if G is quasiprimitive.

It is shown in this paper that the answer to Meagher's question is affirmative for non-quasiprimitive groups of degree 3p, unless possibly when $p=q+1$ is a Fermat prime and Ω admits a unique G-invariant partition $B$ whose blocks are of size 3 such that the induced action $\bar{G}$ is an almost simple group with socle equal to ${\mathrm{PSL}}_2\left(q\right)$.