Intersecting subsets in finite permutation groups

Abstract

Let $G⩽\mathrm{Sym}(\hat{I}\text{c◯})$ be transitive, and let $S$ be an intersecting subset, namely, the ratio $x{y}^{-1}$ of any elements $x,y\in S$ fixes some point. An EKR-type problem is to characterize transitive groups $G⩽\mathrm{Sym}(\hat{I}\text{c◯})$ such that any intersecting set is upper bounded by $|G_{\ddot{I}}‰|$, where $\ddot{I}‰\in \hat{I}\text{c◯}$. A nice result of Meagher-Spiga-Tiep (2016) tells us that if $G$ is 2-transitive, then indeed $|S|⩽|G_{\ddot{I}}‰|$. A natural next step would be to explore intersecting subsets for primitive groups and quasiprimitive groups. Our study in this paper shows that for quasiprimitive permutation groups, the size $|S|$ can be arbitrarily larger than $|G_{\ddot{I}}‰|$. We conjecture that for quasiprimitve groups, the upperbound for $|S|$ is $O(|G_{\ddot{I}}‰||\hat{I}\text{c◯}{|}^{\frac{1}{2}})$. As a starting point, we prove that $|S|/(|G_{\ddot{I}}‰||\hat{I}\text{c◯}{|}^{\frac{1}{2}})⩽\sqrt{2}/2$ for all quasiprimitive actions of the Suzuki groups $G=\mathrm{Sz}(q)$. To show that our conjectured upper bound is tight, we provide examples of groups for which $|S|/(|G_{\ddot{I}}‰||\hat{I}\text{c◯}{|}^{\frac{1}{2}})$ is arbitrarily close to $\sqrt{2}/2$. As far as general transitive groups concerned, infinity families of examples produced show that the ratio $|S|/(|G_{\ddot{I}}‰||\hat{I}\text{c◯}{|}^{\frac{1}{2}})$ can be arbitrarily large.

26 pages, 2 figures 1.Replace the Theorem 1.5 from 'intersecting subgroups of Suzuki groups' to '$\ddot{I}(G/\hat{I}\text{c◯})<1/\sqrt{2}$ when G is a Suzuki group'. 2.Remove Proposition 2.4 3. Add Examples2.5-2.6, Lemma 2.7-2.8