On the Largest intersecting set in $GL_2(q)$ and some of its subgroups

Abstract

Let q be a power of a prime number and V be the 2-dimensional column vector space over a finite field Fq. Assume that SL2(V ) < G ≤ GL2(V ). In this paper we prove an Erdős-Ko-Rado theorem for intersecting sets of G. We show that every maximum intersecting set of G is either a coset of the stabilizer of a point or a coset of H w , whereAlso we get the Hilton-Milner type result for G, i.e. we obtain a bound on the size of the largest intersecting set of G that is neither a coset of the stabilizer of a point nor a coset of H w .

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

“…The following theorem was recently proved by Ahanjideh [1]. It was also partially obtained by Meagher and the second author in [10].…”

“…In [1], it was claimed that if F ⊂ GL 2 (F q ) is an intersecting set which is not contained in a coset of a point stabilizer nor a coset of a stabilizer of a line in O 2 , then |F| ≤ (q − 1)(q − 2) + 1 and this bound is sharp. Unfortunately, we were able to find counterexamples to this claim using Sagemath [11].…”

“…Unfortunately, we were able to find counterexamples to this claim using Sagemath [11]. Note that if such intersecting sets as claimed in [1] exist, then a maximal intersecting set with the aforementioned property will also exist. However, all maximal intersecting sets of GL 2 (F q ), for q ∈ {3, 4, 5}, are all of maximum size; i.e., q(q − 1).…”

“…Moreover, the size of the largest cocliques that are non-canonical are described by the Hilton-Milner theorem [8]. In particular, the maximum cocliques of K(q + 1, 2) have size q and the non-canonical cocliques are of size at most 3 (see (1)).…”

“…Consequently, the graph K(F) is a coclique of K(q + 1, 2). By the Hilton-Milner theorem, we know that if the vertices of F are not contained in any canonical intersecting set of K(q + 1, 2), then |F| ≤ 3 (see (1)). If |F| ≥ 4, then F must be in a canonical intersecting set.…”