Remark on Wielandt's visibility theorem
type: Item
instance of: scholarly article
title: Remark on Wielandt's visibility theorem
zbMATH Open document ID: 0582.20003
DOI: 10.1016/0024-3795(85)90232-0
author: J. J. Seidel
author: Aart Blokhuis
published in: Linear Algebra and its Applications
publication date: 1985-01-01T00:00:00Z
review text: In his classification of permutation groups of degree \(p^ 2\), p a prime number, \textit{H. Wielandt} proved the following theorem [Permutation Groups through Invariant Relations and Invariant Functions, Lect. Notes Ohio State Univ. (1969)]: Let S be a collection of k lines through the origin in the 2-dimensional affine geometry AG(2,p) with p prime and \(1\leq k\leq (p+1)\). Let G be a permutation group acting on the points of AG(2,p) such that G contains all translations of AG(2,p). Then, if G fixes S as a point set, any \(g\in G\) maps any line of S into a line of AG(2,p). In the present note the authors prove that Wielandt's theorem is a consequence of the following theorem by Rédei which was proved in an elementary way by \textit{L. Lovász} and \textit{A. Schrijver} [Stud. Sci. Math. Hung. 16, 449-454 (1981; Zbl 0535.51009)]: Let p be a prime, and let X be a subset of AG(2,p) with \(| X| =p\). Then either X is a line, or X determines at least \((p+3)/2\) directions (of lines intersecting X in at least two points).
Mathematics Subject Classification ID: 20B25
Mathematics Subject Classification ID: 20B10
Mathematics Subject Classification ID: 51E15
zbMATH DE Number: 3931235
zbMATH Keywords: permutation groups of degree \(p^ 2\)
zbMATH Keywords: affine geometry
zbMATH Keywords: translations
MaRDI profile type: Publication
full work available at URL: 0024-3795(85)90232-0
OpenAlex ID: W1973102747
cites work: Q3319331
cites work: Q4083813
description: scientific article; zbMATH DE number 3931235