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The intersection densities of transitive actions of $\operatorname{PSL}_{2}(q)$ with cyclic point stabilizers

Angelot Behajaina, Roghayeh Maleki, Andriaherimanana Sarobidy Razafimahatratra

TL;DR

The paper determines the intersection densities of $\operatorname{PSL}_{2}(q)$ acting transitively on cosets of cyclic subgroups of prime order, focusing first on stabilizers of order $p$ (the characteristic) and then on primes $r$ dividing $(q\pm1)/2$. It employs the derangement graph and its first subconstituent to translate intersecting sets into cliques, exploiting the transitivity of $\operatorname{PGL}_{2}(q)$ on relevant classes. The main results give exact densities: $\rho(\operatorname{PSL}_2(q))=q/p$ for odd $k$ (stabilizer $H_q$) and $\rho(\operatorname{PSL}_2(q))=\sqrt{q}/p$ for even $k$ (stabilizers $H_q^-$ or $H_q^+$). For $r\mid (q\pm1)/2$ with $r$ odd, a sharp lower bound $\rho(G_q)\ge (3r-1)/(2r)$ is obtained via explicit intersecting sets, with the bound realized in many cases, enriching the understanding of intersection densities in transitive groups and their associated arc-transitive graphs.

Abstract

Given a finite transitive group $G\leq \operatorname{Sym}Ω$, the {intersection density} of $G$ is defined as the ratio between the size of the largest subsets of $G$ in which any two permutations agree on at least one element of $Ω$, and the order of a point stabilizer of $G$. In this paper, we completely determine the intersection densities of the permutation groups $\operatorname{PSL}_{2}(q)$, where $q$ is a power of an odd prime $p$, acting transitively with point stabilizers conjugate to $\mathbb{Z}_p$. Our proof uses an auxiliary graph, which is a $\operatorname{PGL}_{2}{q}$-vertex-transitive graph, in which a clique corresponds to an intersecting set of $\operaotnrame{PSL}_{2}(q)$. For the transitive action of $\psl{2}{q}$ with point stabilizers conjugate to $\mathbb{Z}_r$, where $r\mid \frac{q-1}{2}$ is an odd prime, we show that the auxiliary graph is not regular, and we construct an intersecting set which is sometimes of maximum size.

The intersection densities of transitive actions of $\operatorname{PSL}_{2}(q)$ with cyclic point stabilizers

TL;DR

The paper determines the intersection densities of acting transitively on cosets of cyclic subgroups of prime order, focusing first on stabilizers of order (the characteristic) and then on primes dividing . It employs the derangement graph and its first subconstituent to translate intersecting sets into cliques, exploiting the transitivity of on relevant classes. The main results give exact densities: for odd (stabilizer ) and for even (stabilizers or ). For with odd, a sharp lower bound is obtained via explicit intersecting sets, with the bound realized in many cases, enriching the understanding of intersection densities in transitive groups and their associated arc-transitive graphs.

Abstract

Given a finite transitive group , the {intersection density} of is defined as the ratio between the size of the largest subsets of in which any two permutations agree on at least one element of , and the order of a point stabilizer of . In this paper, we completely determine the intersection densities of the permutation groups , where is a power of an odd prime , acting transitively with point stabilizers conjugate to . Our proof uses an auxiliary graph, which is a -vertex-transitive graph, in which a clique corresponds to an intersecting set of . For the transitive action of with point stabilizers conjugate to , where is an odd prime, we show that the auxiliary graph is not regular, and we construct an intersecting set which is sometimes of maximum size.

Paper Structure

This paper contains 19 sections, 17 theorems, 42 equations, 1 table.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main results
  4. Organization of the paper
  5. Background and preliminaries
  6. Graph theory
  7. Elements and subgroups of order $p$
  8. Notations
  9. Subgroups of order $p$
  10. Elements of order $p$
  11. Proof of Theorem \ref{['thm:mainthm1']}
  12. The intersection density of $G_q$ on cosets of $H_q$ ($k$ odd)
  13. Proof strategy for Proposition \ref{['prop:interdensgqhq']}
  14. Proofs of the required lemmas in §\ref{['ss:prstrainter']}
  15. The intersection density of $G_q$ on cosets of $\leftindex^-H_q$ and $\leftindex^{+}{H}_q$ ($k$ even)
  16. ...and 4 more sections

Key Result

Theorem 1.2

If $q = p^k$, where $k\geq 1$ is an integer and $p$ is a prime, then the intersection density of $\operatorname{PSL}_2(q)$ in its action on the cosets of $H = \mathbb{Z}_3$ is given by:

Theorems & Definitions (26)

  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 16 more