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

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 $\mathcal{F}$ of $G$ is an intersecting set if every pair of elements of $\mathcal{F}$ is intersecting. The intersection density $\rho (G)$ of a transitive permutation group $G$ is the maximum value of the quotient $|\mathcal{F}|/|G_v|$ where $G_v$ is a stabilizer of $v\in V$ and $\mathcal{F}$ 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 $p$.

8 pages

References in corpus (1)

• Intersection density of transitive groups of certain degrees