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

Abstract

Given a group $G$ and a subgroup $H\le G$, a set $\mathcal{F}\subset G$ is called $H$\emph{-intersecting} if for any $g,g^{\prime }\in \mathcal{F}$, there exists $xH\in G/H$ such that $gxH=g^{\prime }xH$. The \emph{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 \emph{intersection spectrum} of the group $G$ is then defined to be the set $$\sigma (G):=\left\{\rho (G,H):H\le G\right\}.$$ It was shown by Bardestani and Mallahi-Karai [{\it 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 ${\mathrm{PSL}}_2(q)$. It is shown that $2\in \sigma \left({\mathrm{PSL}}_2(q)\right)$, for any prime power $q\equiv 3(\mathrm{mod}4)$. Moreover, when $q\equiv 1(\mathrm{mod}4)$, it is proved that $\rho ({\mathrm{PSL}}_2(q),H)=1$, for any odd index subgroup $H$ (containing ${\mathbb{F}}_q$) of the Borel subgroup (isomorphic to ${\mathbb{F}}_q⋊{\mathbb{Z}}_{\frac{q-1}{2}}$) consisting of all upper triangular matrices.