Intersection density of transitive groups of certain degrees

Abstract

Two elements g and h of a permutation group G acting on a set V are said to be intersecting if vg = vh for some v ∈ V . More generally, a subset F of G is an intersecting set if every pair of elements of F is intersecting. The intersection density ρ(G) of a transitive permutation group G is the maximum value of the quotient |F|/|Gv | where F runs over all intersecting sets in G and Gv is a stabilizer of v ∈ 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. 1. Introductory remarks For a finite set V let Sym(V ) and Alt(V ) denote the corresponding symmetric group and alternating group on V . (Of course, if |V | = n the standard notations Sn, An apply.) Let G 6 Sym(V ) be a permutation group acting on a set V . Two elements g, h ∈ G are said to be intersecting if v = v for some v ∈ V . Furthermore, a subset F of G is an intersecting set if every pair of elements of F is intersecting. The intersection density ρ(F) of the intersecting set F is defined to be the quotient ρ(F) = |F| maxv∈V |Gv| , and the intersection density ρ(G) of a group G, first defined by Li, Song and Pantangi in [8], is the maximum value of ρ(F) where F runs over all intersecting sets in G, that is, ρ(G) = max{ρ(F) : F ⊆ G,F is intersecting} = max{|F| : F ⊂ G is intersecting} maxv∈V |Gv| . Manuscript received 26th September 2021, accepted 18th November 2021.