Intersection density of imprimitive groups of degree \(pq\) (Q6612118)

Given a finite set \(\Omega\), a subset \({\mathcal F}\) of the symmetric group Sym\( (\Omega)\) is called \textit{intersecting} if \(g^{-1}h\) fixes some element of \(\Omega\) for any elements \(g,\,h \) of \({\mathcal F}\). The maximal size of an intersecting subset of the symmetric group \(S_n\) is \((n-1)!\) and the maximal intersecting subsets are precisely the cosets of the point stabilisers [\textit{P. J. Cameron} and \textit{C. Y. Ku}, Eur. J. Comb. 24, No. 7, 881--890 (2003; Zbl 1026.05001); \textit{G. Kun} and \textit{B. Larose}, Eur. J. Comb. 30, No. 1, 17--29 (2009; Zbl 1207.05143)]. Moving from the full symmetric group to an arbitrary transitive subgroup \(G \leq Sym (\Omega)\), there may be intersecting sets of size larger than \(\vert G_{\omega} \vert\). The \textit{intersection density} \(\rho({\mathcal F})\) of an intersecting subset \({\mathcal F}\) of \(G\) is \(\vert \frac{\vert {\mathcal F} \vert }{\vert G_{\omega} \vert} \) and the intersection density \(\rho(G)\) is \(\max \{ \rho({\mathcal F}) \mid {\mathcal F} \subseteq G \text{ is intersecting}\}\).\N\NThe paper is concerns with studying the intersection density of transitive groups of degree \(pq\), with \(p>q\) and \(p\), \(q\) distinct primes. In [\textit{A. Hujdurović} et al., Finite Fields Appl. 78, Article ID 101975, 10 p. (2022; Zbl 1514.20016)] a family of transitive imprimitive groups of degree \(pq\) was constructed whose intersection densities are equal to \(q\). The authors conjecture that \(\rho(G) = 1\) for any transitive quasiprimitive group of degree \(pq\) and prove this for a certain family of such groups, namely the groups \(PSL_q(q)\) where \(p = \frac{q^q-1}{q-1}\) is a prime (Conjecture \(1.3\) and Theorem \(1.2\), respectively). As to imprimitive groups of degree \(pq\), Theorem \(1.4\) provides a set of sufficient conditions for \(\rho(G) = 1\), while Theorem \(1.7\) shows that, if \(q\) and \(p = \frac{q^k-1}{q-1}\) are odd primes, then \(\frac{q}{k}\) occurs as intersection density of an appropriate (imprimitive) group of degree \(pq\).