Biplanes with flag-transitive automorphism groups of almost simple type, with alternating or sporadic socle.
type: Item
instance of: scholarly article
title: Biplanes with flag-transitive automorphism groups of almost simple type, with alternating or sporadic socle.
zbMATH Open document ID: 1113.20003
published in: European Journal of Combinatorics
publication date: 2005-05-04T00:00:00Z
review text: Biplanes are a generalization of projective planes in the sense that it is assumed that there are exactly two lines through any two points; in design theory language, a biplane is a \((v,k,2)\)-symmetric design. Very few are known, there are none known for \(k>13\). This paper contributes to the problem by considering biplanes with flag-transitive automorphism groups. The corollary of the main theorem is: Let \(D\) be a biplane with a flag-transitive automorphism group \(G\), then one of the following holds: (1) \(D\) has parameters \((16,6,2)\), (2) \(G\leq A\Gamma L_1(q)\) for some odd prime power, or (3) \(G\) is almost simple with socle a Chevalley group. The proof depends on the fact that we have a complete description of the maximal subgroups and hence we can identify the possible stabilizers of points quite easily. In the sporadic case the results come from knowing the maximal subgroups and largely using numerology to show that there is no such action.
reviewed by: A. R. Camina
Mathematics Subject Classification ID: 20B25
Mathematics Subject Classification ID: 05B05
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 20B15
zbMATH DE Number: 2165401
zbMATH Keywords: flag-transitive automorphism groups
zbMATH Keywords: symmetric block designs
zbMATH Keywords: biplanes
zbMATH Keywords: alternating groups
zbMATH Keywords: sporadic groups
author: Eugenia O'Reilly-Regueiro
describes a project that uses: GAP
MaRDI profile type: Publication
full work available at URL: j.ejc.2004.05.003
OpenAlex ID: W2067853228
cites work: On collineation groups of symmetric block designs
cites work: Q4133993
cites work: Biplanes
cites work: Q3684278
cites work: Flag-transitivity and primitivity
cites work: Q2724738
cites work: Q3885773
cites work: Designs derived from permutation groups
cites work: Q5836508
cites work: Q4256627
cites work: The Maximal Subgroups of J 4
cites work: The maximal subgroups of Fi22
cites work: Q3791327
cites work: The Maximal Subgroups of the Thompson Group
cites work: On primitivity and reduction for flag-transitive symmetric designs
cites work: The four biplanes with k=9
cites work: Q5792694
cites work: Q5512231
DOI: 10.1016/J.EJC.2004.05.003
Recommended article: Biplanes with flag-transitive automorphism groups of almost simple type, with classical socle
Recommended article: Biplanes with flag-transitive automorphism groups of almost simple type, with exceptional socle of Lie type.
Recommended article: Q5134916
Recommended article: Characterization of biplanes by their automorphism groups
Recommended article: Finite flag-transitive affine planes with a solvable automorphism group
Recommended article: On biplanes (56, 11, 2) with automorphism groups of order four
Recommended article: On automorphism groups of a biplane \((121,16,2)\)
Recommended article: Alternating groups and flag-transitive triplanes
Recommended article: Biplanes (56,11,2) with a fixed-point-free involutory automorphism
description: scientific article; zbMATH DE number 2165401