The Erdős-Ko-Rado Property for Some 2-Transitive Groups

Abstract

A subset S of a group G ≤ Sym(n) is intersecting if for any pair of permutations π, σ ∈ S there is an i ∈ {1, 2, . . . , n} such that π(i) = σ (i). It has been shown, using an algebraic approach, that the largest intersecting sets in each of Sym(n), Alt(n), and PGL(2, q) are exactly the cosets of the point-stabilizers. In this paper, we show how this approach can be applied more generally to many 2-transitive groups. We then apply this method to the Mathieu groups and to all 2-transitive groups with degree no more than 20.

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

“…It is shown in [17] that the Suzuki groups, Ree Groups, Higman-Sims, Symplectic groups also have the EKR-module property. The result in this paper shows that every minimal almost-simple 2-transitive groups (except possibly PSU (3,5)) has the EKR-module property. Thus we conclude with the following conjecture.…”

“…It is noted in [2] that for any 2-transitive group, the character χ(g) = fix(g) − 1 is an irreducible character (throughout this section we will use χ to denote this character). Then, for any group with a 2-transitive action on a set of size n that has exactly D derangements, the eigenvalue of the derangement graph corresponding to this irreducible character is −|D|/(n − 1) (see [2] for details).…”

“…It is noted in [2] that for any 2-transitive group, the character χ(g) = fix(g) − 1 is an irreducible character (throughout this section we will use χ to denote this character). Then, for any group with a 2-transitive action on a set of size n that has exactly D derangements, the eigenvalue of the derangement graph corresponding to this irreducible character is −|D|/(n − 1) (see [2] for details). For many groups (for example the symmetric group, the alternating group and PGL(2, q)), this is the least eigenvalue of the derangement graph, and Hoffman's ratio bound immediate shows that these groups have the EKR property (showing that the groups have strict-EKR property is where the work lies).…”

“…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].…”

“…For example, the method described in [2] proves that a group with the EKRmodule property has the strict-EKR property, provided that a certain matrix has full rank. For q = 2 this matrix does not have full rank.…”