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 ∈ V . More generally, a subset F of G is an intersecting set if every pair of elements of F is intersecting. The intersection density ρ(G) of a transitive permutation group G is the maximum value of the quotient |F |/|G v | where G v is a stabilizer of v ∈ V and 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 F q whose codewords have Hamming weight strictly smaller than p.