On collineation groups with block orbits
type: Item
instance of: scholarly article
title: On collineation groups with block orbits
zbMATH Open document ID: 0786.51002
DOI: 10.1016/0097-3165(93)90092-M
author: Yutaka Hiramine
published in: Journal of Combinatorial Theory. Series A
publication date: 1993-12-02T00:00:00Z
review text: During the 1960's and 1970's finite affine planes \(\Pi\) having collineation groups \(G\) transitive on the affine points were investigated, and theorems characterizing such planes were obtained. In many cases -- for example, planes with groups doubly transitive on the affine points -- the crux of the matter was the fact that the subgroup \(G_ 0\) fixing an affine point 0 has a block orbit; i.e., a point orbit \(\Gamma\) such that \(\Gamma \cup \{0\}\) is the union of one or more affine lines through 0 [see the reviewer's book `Affine planes with transitive collineation groups' (1982; Zbl 0485.51006)]. In that text the reviewer made the conjecture that if \(G_ 0\) has a block orbit \(\Gamma\) with \(\Gamma \cup \{0\}\) the union of at least two lines, then \(\Pi\) is a translation plane and \(G\) contains the group of translation. \textit{N. L. Johnson} and the reviewer [Math. Z. 135, 149-164 (1974; Zbl 0274.50017)] showed that the conjecture is true under any one of additional conditions -- for example, if the plane has nonsquare order, or if its order is even. In the present article the author shows that the conjecture is true without any additional hypothesis. The reviewer also made two other conjectures: (i) If \(\Gamma \cup \{0\}\) is an affine line through 0 then \(\Pi\) is either a translation plane or a dual translation plane, and the group \(G\) contains the translations or dual translations, respectively. (ii) If \(G_ 0\) has two block orbits then \(\Pi\) is a translation plane and \(G\) contains the translations. The author gives an example of a collineation group \(G\) on a desarguesian plane which satisfies the hypothesis of (i) and (ii), but does not satisfy the conclusions regarding \(G\). This example also disproves a conjecture of Johnson that every block orbit is a strong block orbit. Note, however, that the example does not disprove the conclusions in (i) and (ii) regarding the plane \(\Pi\), but it makes them less likely to be true.
Mathematics Subject Classification ID: 51A40
Mathematics Subject Classification ID: 51E15
Mathematics Subject Classification ID: 20B20
zbMATH DE Number: 446860
zbMATH Keywords: block orbits
zbMATH Keywords: collineation groups
zbMATH Keywords: translation plane
reviewed by: Michael J. Kallaher
MaRDI profile type: Publication
cites work: Q5577154
cites work: On finite affine planes with a 2-transitive orbit on \(\ell_ \infty\)
cites work: Q5683869
cites work: Q3945140
description: scientific article; zbMATH DE number 446860