On the intersection density of primitive groups of degree a product of two odd primes

Abstract

A subset F of a finite transitive group G≤Sym(Ω) is intersecting if for any g,h∈F there exists ω∈Ω such that ωg=ωh. The intersection densityρ(G) of G is the maximum of {|F||Gω||F⊂G is intersecting}, where Gω is the stabilizer of ω in G. In this paper, it is proved that if G is an imprimitive group of degree pq, where p and q are distinct odd primes, with at least two systems of imprimitivity then ρ(G)=1. Moreover, if G is primitive of degree pq, where p and q are distinct odd primes, then it is proved that ρ(G)=1, whenever the socle of G admits an imprimitive subgroup.