Primitive arcs in \(\mathrm{PG}(2,q)\)
type: Item
instance of: scholarly article
title: Primitive arcs in \(\mathrm{PG}(2,q)\)
zbMATH Open document ID: 0818.51003
DOI: 10.1016/0097-3165(95)90051-9
published in: Journal of Combinatorial Theory. Series A
publication date: 1995-08-15T00:00:00Z
review text: A \(k\)-arc \(K\) in a finite projective plane is a set of \(k\) points, no three of which are collinear. A \(k\)-arc is said to be complete if it cannot be included in a \(k+1\) arc of the same plane. This paper gives the classification of all complete \(k\)-arcs in \(\mathrm{PG}(2,q)\) which are invariant under a transitive projective group active primitively on the points of the arc. Let \(G\) be the maximum subgroup \(\mathrm{PGL}(3,q)\) which leaves \(K\) invariant. There are two main cases: I. \(G\) contains an elementary abelian group which acts regularly on \(K\). II. \(G\) is an almost simple group with socle \(S\), where \(S\) is nonabelian simple; \(G \subseteq \Aut S\). There are three types of possibilities for \(S\), namely \(L_3(q')\), \(U_3(q')\) and \(L_2(q')\) for suitable \(q'\) dividing \(q\). Most of the arcs are conics in \(\mathrm{PG}(2,q)\), conics in some subplane, hyperovals, orbits under a Singer group or are sporadic cases with relatively small \(k\).
Mathematics Subject Classification ID: 51E21
Mathematics Subject Classification ID: 20B25
zbMATH DE Number: 734533
zbMATH Keywords: \(k\)-arcs
author: Leo Storme
author: Hendrik Van Maldeghem
reviewed by: Theodore G. Ostrom
MaRDI profile type: Publication
full work available at URL: 0097-3165(95)90051-9
OpenAlex ID: W2000798455
cites work: The Subgroups of PSL(3, q) for odd q
cites work: Q3798841
cites work: Q3684278
cites work: Q5545191
cites work: Hexagons, Conics, A5 and PSL2 (K )
cites work: Q3851217
cites work: Q3694152
cites work: Q4003762
cites work: Cyclic arcs in \(\mathrm{PG}(2,q)\)
cites work: Q4882705
cites work: Finite groups in which the centralizer of any element of order 2 is 2- closed
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description: scientific article; zbMATH DE number 734533