Intersection density of imprimitive groups of degree pq

A subset $F$ of a finite transitive group $G\le \mathrm{Sym}\left(\Omega \right)$ is intersecting if any two elements of $F$ agree on an element of Ω. The intersection density of G is the number$\rho \left(G\right)=\max \{\left|F\right|/|G_{\omega }\left|\right|F\subset G\text{ is intersecting}\},$ where $\omega \in \Omega$ and $G_{\omega }$ is the stabilizer of ω in G. It is known that if $G\le \mathrm{Sym}\left(\Omega \right)$ is an imprimitive group of degree a product of two odd primes $p>q$ admitting a block of size p or two complete block systems, whose blocks are of size q, then $\rho \left(G\right)=1$.

In this paper, we analyze the intersection density of imprimitive groups of degree pq with a unique block system with blocks of size q based on the kernel of the induced action on blocks. For those whose kernels are non-trivial, it is proved that the intersection density is larger than 1 whenever there exists a cyclic code C with parameters ${[p,k]}_q$ such that any codeword of C has weight at most $p-1$, and under some additional conditions on the cyclic code, it is a proper rational number. For those that are quasiprimitive, we reduce the cases to almost simple groups containing $\mathrm{Alt}\left(5\right)$ or a projective special linear group. We give some examples where the latter has intersection density equal to 1, under some restrictions on p and q.