On the Erdős–Ko–Rado property for finite groups

Abstract

Let a finite group G act transitively on a finite set X . A subset S ⊆ G is said to be intersecting if for any s 1 , s 2 ∈ S, the element s −1 1 s 2 has a fixed point. The action is said to have the weak Erdős-Ko-Rado (EKR) property, if the cardinality of any intersecting set is at most |G|/|X |. If, moreover, any maximum intersecting set is a coset of a point stabilizer, the action is said to have the strong EKR property. In this paper, we will investigate the weak and strong EKR property and attempt to classify groups in which all transitive actions have these properties. In particular, we show that a group with the weak EKR property is solvable and that a nilpotent group with the strong EKR property is the direct product of a 2-group and an abelian group of odd order.

Some Erdös-Ko-Rado results for linear and affine groups of degree two

“…Since any group action, is equivalent to the action of the group on a set of cosets, this is a relatively tractable problem. Most research on EKR-type results for groups focuses on well-known group actions, there is a growing body of work considering all the actions of a group [8,23]. Since the character table GL(2, q) is completely understood, it should be straightforward to try this method for different actions of the general linear group.…”

Erdős-Ko-Rado results for the general linear group, the special linear group and the affine general linear group

“…Since any group action, is equivalent to the action of the group on a set of cosets, this is a relativey tractable problem. Most research on EKR-type results for groups focuses on well-known group actions, there is a growing body of work considering all the actions of a group [8,21]. Since the character table GL(2, q) is completely understood, it should be straightforward to try this method for different actions of the general linear group.…”

On the EKR-module property

“…It is shown in [7,Theorem 3] that there are infinitely many nilpotent groups of nilpotency class 2 that do not satisfy the strict-EKR property. The next result then answers the question in negative.…”