No Hilton-Milner type results for linear groups of degree two

2021/11/06 by Roghayeh Maleki, Maleki, Roghayeh, Andriaherimanana Sarobidy Razafimahatratra +1
Engineering · Mathematics · Medicine · #Chronic Lymphocytic Leukemia Research #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2111.03829

openalex publication_date 2021/11/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A set of permutations F of a finite transitive permutation group G≤ Sym(Ω) is intersecting if any pair of elements of F agree on an element of Ω. We say that G has the EKR property if an intersecting set of G has size at most the order of a point stabilizer. Moreover, G has the strict-EKR property whenever G has the EKR property and any intersecting set of maximum size is a coset of a point stabilizer of G. It is known that the permutation group GL₂(\mathbbF_q) acting on Ω_q := \mathbbF_q²∖\0\ has the EKR property, but does not have the strict-EKR property since the stabilizer of a hyperplane is a maximum intersecting set. In this paper, it is proved that the Hilton-Milner type result does not hold for GL₂(\mathbbF_q) acting on Ω_q. Precisely, it is shown that a maximal intersecting set of GL₂(\mathbbF_q) is of maximum size. As a result, we prove the Complete Erdős-Ko-Rado theorem for GL₂(\mathbbF_q).

Related

• The Erdős-Ko-Rado property for some permutation groups
• Maximum Intersecting Families of Permutations
• The Erdős-Ko-Rado property for some 2-transitive groups
• Erd\Hos-Ko-Rado problems for permutation groups
• Intersection density of transitive groups with cyclic point stabilizers