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 densityρ(G) of a transitive permutation group G is the maximum value of the quotient |F|/|Gv| where Gv is the 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 Meagher et al. (2021) [15]) is disproved by constructing a family of imprimitive permutation groups of degree pq (with blocks of size q), where p=(qk−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 Fq whose codewords have Hamming weight strictly smaller than p.