Linear subspaces of finite fields with large inverse-closed subsets
type: Item
instance of: scholarly article
title: Linear subspaces of finite fields with large inverse-closed subsets
zbMATH Open document ID: 1275.12001
author: Bence Csajbók
published in: Finite Fields and their Applications
publication date: 2013-01-14T00:00:00Z
review text: Let \(F\) be a finite field (or an infinite field with characteristic distinct from 2) and let \(A\) be a nontrivial subgroup of \((F,+)\) such that \(\{ a^{-1} \mid 0 \neq a \in F \} \cup \{0\}\) is also an additive subgroup. The author proves that \(A\) is then a one-dimensional subspace over a subfield of \(F\). This is a corollary of more general results which say that an additive subgroup of \(F\) with a large inverse-closed subset is necessarily inverse-closed. Examples from finite geometry show that some bounds in these results are sharp. Related results are contained \textit{S. Mattarei} [Isr. J. Math. 159, 343--347 (2007; Zbl 1211.12007)], \textit{D. Goldstein} et al. [Pac. J. Math. 227, No. 2, 287--294 (2006; Zbl 1122.17024)] and \textit{H.-J. Kroll} [Geom. Dedicata 43, No. 1, 59--64 (1992; Zbl 0758.51004)].
Mathematics Subject Classification ID: 12E20
Mathematics Subject Classification ID: 51E22
zbMATH DE Number: 6125338
zbMATH Keywords: finite field
zbMATH Keywords: inverse-closed
zbMATH Keywords: cap
zbMATH Keywords: arc
reviewed by: Theo Grundhöfer
MaRDI profile type: Publication
full work available at URL: j.ffa.2012.10.005
OpenAlex ID: W2064770353
DOI: 10.1016/J.FFA.2012.10.005
description: scientific article; zbMATH DE number 6125338