On intersection density of transitive groups of degree a product of two odd primes

TL;DR

This paper investigates the intersection density of transitive groups of degree pq, disproving a conjecture by constructing imprimitive groups with intersection density q, using properties of equidistant cyclic codes over finite fields.

Contribution

It provides a counterexample to the conjecture that all such groups have intersection density 1, by constructing specific imprimitive groups with higher density based on cyclic codes.

Findings

Disproved the conjecture that intersection density is always 1 for degree pq groups.

Constructed imprimitive groups with intersection density q.

Linked group properties to cyclic code structures over finite fields.

Abstract

Two elements $g$ and $h$ of a permutation group $G$ acting on a set $V$ are said to be intersecting if $g(v) = h(v)$ for some $v \in V$. More generally, a subset ${\cal F}$ of $G$ is an intersecting set if every pair of elements of ${\cal F}$ is intersecting. The intersection density $\rho(G)$ of a transitive permutation group $G$ is the maximum value of the quotient $|{\cal F}|/|G_v|$ where $G_v$ is a stabilizer of $v\in V$ and ${\cal F}$ runs over all intersecting sets in $G$. Intersection densities of transitive groups of degree $pq$, where $p>q$ are odd primes, is considered. In particular, the conjecture that the intersection density of every such group is equal to $1$ (posed in [ J.~Combin. Theory, Ser. A 180 (2021), 105390]) is disproved by constructing a family of imprimitive permutation groups of degree $pq$ (with blocks of size $q$), where $p=(q^k-1)/(q-1)$, whose intersection density is equal to $q$. The construction depends heavily on certain equidistant cyclic codes $[p,k]_q$ over the field $\mathbb{F}_q$ whose codewords have Hamming weight strictly smaller than $p$.