On the intersection spectrum of $${\text {PSL}}_2(q)$$

Abstract

Given a group G and a subgroup \(H \le G\), a set \(\mathcal {F}\subset G\) is called H-intersecting if for any \(g,g' \in \mathcal {F}\), there exists \(xH \in G/H\) such that \(gxH=g'xH\). The intersection density of the action of G on G/H by (left) multiplication is the rational number \(\rho (G,H)\), equal to the maximum ratio \(\frac{|\mathcal {F}|}{|H|}\), where \(\mathcal {F} \subset G\) runs through all H-intersecting sets of G. The intersection spectrum of the group G is then defined to be the set

$$\begin{aligned} \sigma (G) := \left\{ \rho (G,H) : H\le G \right\} . \end{aligned}$$

It was shown by Bardestani and Mallahi-Karai (J Algebraic Combin, 42(1):111–128, 2015) that if \(\sigma (G) = \{1\}\), then G is necessarily solvable. The natural question that arises is, therefore, which rational numbers larger than 1 belong to \(\sigma (G)\), whenever G is non-solvable. In this paper, we study the intersection spectrum of the linear group \({\text {PSL}}_2(q)\). It is shown that \(2 \in \sigma \left( {\text {PSL}}_2(q)\right) \), for any prime power \(q\equiv 3 \pmod 4\). Moreover, when \(q\equiv 1 \pmod 4\), it is proved that \(\rho ({\text {PSL}}_2(q),H)=1\), for any odd index subgroup H (containing \({\mathbb {F}}_q\)) of the Borel subgroup (isomorphic to \({\mathbb {F}}_q\rtimes {\mathbb {Z}}_{\frac{q-1}{2}}\)) consisting of all upper triangular matrices.