An Erdös--Ko--Rado Theorem for the Derangement Graph of ${PGL}_3(q)$ Acting on the Projective Plane

Abstract

In this paper we prove an Erdős-Ko-Rado-type theorem for intersecting sets of permutations. We show that an intersecting set of maximal size in the projective general linear group PGL 3 (q), in its natural action on the points of the projective line, is either a coset of the stabilizer of a point or a coset of the stabilizer of a line. This gives the first evidence to the veracity of [11, Conjecture 2].

An Erdős-Ko-Rado theorem for the group $$\hbox {PSU}(3, q)$$ PSU ( 3 , q )

“…Namely, we prove that the characteristic vector for any maximum intersecting set in PSU(3, q) is a linear combination of characteristic vectors of the canonical cliques in PSU (3, q). Showing the same result for other 2-transitive groups was a key result in proving that the groups have the strict-EKR property [1,10,15,16].…”

“…The previous lemma implies that if G has the EKR-module property, then any maximum coclique in such a Γ G is a linear combination of the characteristic vectors of the canonical cocliques. The symmetric and alternating group has the EKR-module property [1,10], as does PGL(2, q), PGL(3, q) and the Mathieu groups [2,15,16].…”

“…Although all 2-transitive groups have the ERK-property, it is not true that all 2-transitive groups have the strict-EKR property. It has been shown that Sym(n), Alt(n), PGL(2, q), PGL(3, q), PSL(2, q) [14] and the Mathieu groups all have the strict-EKR property by first showing that they have the EKR-module property [1,2,10,15,16,14]. It is shown in [17] that the Suzuki groups, Ree Groups, Higman-Sims, Symplectic groups also have the EKR-module property.…”

On the Erdős–Ko–Rado property for finite groups

“…Nevertheless, it may still have a chance to satisfy a weaker variation of it. In [9], Meagher and Spiga conjecture that the only intersecting sets of the maximum size are the cosets of point and hyperplane stabilizers. This conjecture has been verified for the case n = 3 by the same authors in [10].…”

“…In [9], Meagher and Spiga conjecture that the only intersecting sets of the maximum size are the cosets of point and hyperplane stabilizers. This conjecture has been verified for the case n = 3 by the same authors in [10]. Similarly, using elementary arguments, we can prove the following result for a similar action: This paper is organized as follows.…”