An Erdős-Ko-Rado theorem in general linear groups

2011/07/15 by Jun Guo, Guo, Jun, Kaishun Wang +1
Computer Science · Mathematics · #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Limits and Structures in Graph Theory

paper · pdf · doi:10.48550/arxiv.1107.3178

openalex publication_date 2011/07/15 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

Let S_n be the symmetric group on n points. Deza and Frankl [M. Deza and P. Frankl, On the maximum number of permutations with given maximal or minimal distance, J. Combin. Theory Ser. A 22 (1977) 352--360] proved that if \cal F is an intersecting set in S_n then |\cal F|≤(n-1)!. In this paper we consider the q-analogue version of this result. Let \mathbbF_qⁿ be the n-dimensional row vector space over a finite field \mathbbF_q and GL_n(\mathbbF_q) the general linear group of degree n. A set \cal F_q⊆ GL_n(\mathbbF_q) is \it intersecting if for any T,S∈\cal F_q there exists a non-zero vector α∈ \mathbbF_qⁿ such that αT=αS. Let \cal F_q be an intersecting set in GL_n(\mathbbF_q). We show that |\cal F_q|≤ q^(n-1)n/2∏_i=1^n-1(qⁱ-1).

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