Inequalities involving the Delandtsheer--Doyen parameters for finite line-transitive linear spaces
type: Item
instance of: scholarly article
title: Inequalities involving the Delandtsheer--Doyen parameters for finite line-transitive linear spaces
zbMATH Open document ID: 1026.05019
DOI: 10.1016/S0097-3165(03)00012-8
author: Cheryl E. Praeger
author: Ngo Dac Tuan
published in: Journal of Combinatorial Theory. Series A
publication date: 2003-05-27T00:00:00Z
review text: Let \(\mathcal{L}\) be a linear space with a line-transitive, point-imprimitive automorphism group \(G\). In [\textit{A. Delandtsheer} and \textit{J. Doyen}, Geom. Dedicata 29, 307-310 (1989; Zbl 0673.05010)] two integer parameters \(x\) and \(y\) were introduced for every \(G\)-invariant nontrivial partition of the point set. The paper explores inequalities involving these two parameters. There is special interest to the case where the \(G\)-invariant partition \(\mathcal{C}\) is \(G\)-normal, that is, to the case where \(\mathcal{C}\) is the set of orbits of a normal subgroup \(K\) of \(G\). If the constant line size is greater than \(2x+\frac{3}{2}+\sqrt{4x-\frac{7}{4}}\), then the authors show that \(K\) acts semiregularly on points and on lines. If the parameter \(x\) is at most 8, then it is shown that either \(K\) is abelian and semiregular on points or \(\mathcal{L}\) is one of four explicitly known examples.
reviewed by: Bart De Bruyn
Mathematics Subject Classification ID: 05B25
Mathematics Subject Classification ID: 20B25
Mathematics Subject Classification ID: 51E26
Mathematics Subject Classification ID: 51A45
zbMATH DE Number: 1917907
zbMATH Keywords: linear spaces
zbMATH Keywords: automorphism group
Wikidata QID: Q56987893
MaRDI profile type: Publication
cites work: Q5488375
cites work: On a theorem of Wielandt for finite primitive permutation groups
cites work: On the orbits of collineation groups
cites work: Line-transitive automorphism groups of linear spaces
cites work: Line-Transitive Automorphism Groups of Linear Spaces
cites work: Line-transitive, point quasiprimitive automorphism groups of finite linear spaces are affine or almost simple.
cites work: Q3903016
cites work: Most block-transitive t-designs are point-primitive
cites work: Finite line-transitive linear spaces: parameters and normal point-partitions
cites work: Q4882944
cites work: Geometric ABA-groups
cites work: Cyclic 2-(91,6,1) designs with multiplier automorphisms
cites work: Block-transitive, point-imprimitive designs with \(\lambda \Relbar 1\)
cites work: Q4550677
cites work: The characterization of finite groups with abelian Sylow 2-subgroups
cites work: Q5512231
full work available at URL: s0097-3165(03)00012-8
OpenAlex ID: W2045968061
description: scientific article; zbMATH DE number 1917907