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

Abstract

In this paper we consider the derangement graph for the group PSU(3, q) where q is a prime power. We calculate all eigenvalues for this derangement graph and use these eigenvalues to prove that PSU(3, q) has the Erdős-Ko-Rado property and, provided that q = 2, 5, another property that we call the Erdős-Ko-Rado module property.2010 Mathematics Subject Classification. Primary 05C35; Secondary 05C69, 20B05.

The EKR-Module Property of Pseudo-Paley Graphs of Square Order

“…Once we define canonical cocliques (or cliques), we can discuss the EKR properties of G after identifying G with Γ(G). The EKR-module property was first formally defined by Meagher [21] in this context: a permutation group G naturally acts on the vector space W spanned by the characteristic vectors of canonical cliques, which makes W a G-module.…”

On the EKR-module property

“…A transitive group is said to satisfy the EKR-module property if the characteristic vector of every maximum intersecting set is a linear combination of the characteristic vectors of cosets of stabilizers of points. This generalization of the wellknown permutation group version of the Erdős-Ko-Rado (EKR) theorem was introduced by K. Meagher in [28]. In this article, we present several infinite families of permutation groups satisfying the EKR-module property, which shows that permutation groups satisfying this property are quite diverse.…”

“…the characteristic vector of the canonical intersecting set gG α , which we call a canonical vector for convenience. The next definition was first introduced in [28].…”

On the Largest intersecting set in GL 2 (q) and some of its subgroups

“…Finally, Spiga showed that the maximum intersecting sets of PGL n+1 (q) acting on the points of the projective space P n q are the cosets of the stabilizer of a point and the cosets of the stabilizer of a hyperplan [15]. For more studies, we refer the reader to [7,11,14].…”