Remark on Wielandt's visibility theorem
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).
scientific article; zbMATH DE number 3931235