Projective planes and dihedral groups
type: Item
instance of: scholarly article
title: Projective planes and dihedral groups
zbMATH Open document ID: 0902.51003
DOI: 10.1016/S0012-365X(97)80336-6
author: Johannes Ueberberg
published in: Discrete Mathematics
publication date: 1998-12-10T00:00:00Z
review text: Let \(q\) be a prime power and \(P\) the projective plane of order \(q^2\), \(B_0\), \(B_1\) two disjoint Baer subplanes of \(P\), \(\tau_0\), \(\tau_1\) the corresponding Baer involutions, \(\delta =\tau_0 \tau_1\). The author's main result is the following (a) \(\delta\) is a fixed point free projective collineation whose order \(r\) divides \(q^2- q+1\). (b) \(\delta= \sigma^n\) for some Singer cycle \(\sigma\) and some natural number \(n\). (c) \({\mathcal P}= \{\delta (B_0)^j \mid j\in \{0, \dots, r-1\}\}\) is a set of mutually disjoint Baer subplanes. If furthermore \(r=q^2 -q+1\) then \({\mathcal P}\) contains \(B_0\) and \(B_1\) and it coincides with the linear partition induced by \(\sigma\), cf. Theorem 4.3.5 in \textit{J. W. P. Hirschfeld} [`Projective geometry over finite fields' (1979; Zbl 0418.51002)]. The point orbits under the action of \(\langle \delta \rangle\) are complete \((q^2- q+1)\)-arcs of a special form described by \textit{B. C. Kestenband} [Geom. Dedicata 11, 107-117 (1981; Zbl 0452.51007)]. (d) \(D= \langle \tau_0, \tau_1 \rangle\) is a dihedral group of order \(2r\) whose reflections are the Baer involutions arising from the Baer subplanes in \({\mathcal P}\).
Mathematics Subject Classification ID: 51E15
Mathematics Subject Classification ID: 51E23
Mathematics Subject Classification ID: 51C05
Mathematics Subject Classification ID: 51E25
Mathematics Subject Classification ID: 51E20
zbMATH DE Number: 1110040
reviewed by: Horst Szambien
MaRDI profile type: Publication
cites work: On the sharpness of a theorem of B. Segre
cites work: Q3889686
cites work: Q3274609
cites work: Q5606475
cites work: Baer Subplanes in Finite Projective and Affine Planes
cites work: A new proof of the existence of \((q^ 2-q+1)\)-arcs in \(PG(2,q^ 2)\)
cites work: Caps on elliptic quadrics
cites work: Q5545191
cites work: Partitioning Projective Geometries into Caps
cites work: Q3717726
cites work: Q3851217
cites work: Projective Geometries that are Disjoint Unions of Caps
cites work: Unital intersections in finite projective planes
cites work: A family of complete arcs in finite projective planes
cites work: Q5589782
cites work: Q3041885
cites work: On regular {v, n}‐arcs in finite projective spaces
cites work: A note on the intersection of two Baer subplanes
description: scientific article; zbMATH DE number 1110040