Intersection density of transitive groups of certain degrees

TL;DR

This paper determines the intersection density of transitive groups of degrees twice a prime and prime power, showing it is either 1 or 2, and provides new insights into the structure of intersecting sets in permutation groups.

Contribution

It establishes the exact intersection density for transitive groups of degree twice a prime and proves it is 1 for prime power degrees, advancing understanding of intersecting sets.

Findings

Intersection density of degree twice a prime is 1 or 2.

Intersection density of prime power degree is 1.

Provides classification of intersecting sets in these groups.

Abstract

Two elements $g$ and $h$ of a permutation group $G$ acting on a set $V$ are said to be intersecting if $g(v) = h(v)$ for some $v \in V$. More generally, a subset ${\cal F}$ of $G$ is an intersecting set if every pair of elements of ${\cal F}$ is intersecting. The intersection density $\rho(G)$ of a transitive permutation group $G$ is the maximum value of the quotient $|{\cal F}|/|G_v|$ where ${\cal F}$ runs over all intersecting sets in $G$ and $G_v$ is a stabilizer of $v\in V$. In this paper the intersection density of transitive groups of degree twice a prime is determined, and proved to be either $1$ or $2$. In addition, it is proved that the intersection density of transitive groups of prime power degree is $1$.