Finite linear spaces with flag-transitive groups
type: Item
instance of: scholarly article
title: Finite linear spaces with flag-transitive groups
zbMATH Open document ID: 0658.20001
DOI: 10.1016/0097-3165(88)90056-8
author: Francis Buekenhout
author: Anne Delandtsheer
author: Jean Doyen
published in: Journal of Combinatorial Theory. Series A
publication date: 1988-01-01T00:00:00Z
review text: Ein linearer Raum S ist eine Inzidenzstruktur, in der durch zwei verschiedene Punkte genau eine Gerade geht, jede Gerade mindestens zwei Punkte enthält und durch jeden Punkt mindestens zwei Geraden gehen. Eine Automorphismengruppe G auf S ist fahnentransitiv (flag-transitive) auf S, wenn sie transitiv auf allen Fahnen (d.h. inzidenten Punkt- Geraden-Paaren) operiert. Verff. untersuchen die Struktur endlicher fahnentransitiver Räume (S,G). Jeder solche Raum ist ein 2-(v,k,1)- design mit \(k<v\). Zunächst wird ein sehr umfassender historischer Überblick mit Beispielen gegeben. Ist (S,G) fahnentransitiv, so operiert G primitiv auf der Punktmenge von S [\textit{D. G. Higman}, \textit{J. E. McLaughlin}, Ill. J. Math. 5, 382-397 (1961; Zbl 0104.147)]. Damit kann das O'Nan-Scott-Theorem über die Klassifizierung endlicher primitiver Gruppen angewendet werden [vgl. etwa \textit{F. Buekenhout}: Bull. Soc. Math. Belg., Sér. B 40, 1-9 (1988; Zbl 0653.20003)], durch das die Klasse aller endlichen primitiven Gruppen in fünf paarweise disjunkte Klassen zerlegt werden kann: die einfache Klasse, die cartesische Klasse, die affine Klasse, die bireguläre und die diagonale Klasse. Abgesehen von fünf Ausnahmegruppen, die alle auf AG(2,8) operieren, gehören alle G in fahnentransitiven (S,G) zur einfachen oder affinen Klasse.
Mathematics Subject Classification ID: 20B05
Mathematics Subject Classification ID: 20B15
Mathematics Subject Classification ID: 05B25
Mathematics Subject Classification ID: 20B25
Mathematics Subject Classification ID: 05B30
Mathematics Subject Classification ID: 51E20
zbMATH DE Number: 4075367
zbMATH Keywords: finite linear spaces
zbMATH Keywords: flag-transitive automorphism group
zbMATH Keywords: flag-transitive finite linear spaces
zbMATH Keywords: O'Nan-Scott theorem
zbMATH Keywords: primitive permutation groups
zbMATH Keywords: affine type
zbMATH Keywords: simple type
zbMATH Keywords: incidence structure
zbMATH Keywords: 2-(v,k,1)-design
reviewed by: Johannes André
MaRDI profile type: Publication
full work available at URL: 0097-3165(88)90056-8
OpenAlex ID: W1971802219
cites work: Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe
cites work: Projektive Ebenen über Fastkörpern
cites work: Polarities in finite projective planes
cites work: Q3721311
cites work: Construction of two-dimensional flag-transitive planes
cites work: On the orbits of collineation groups
cites work: On the Construction of Sets of Mutually Orthogonal Latin Squares and the Falsity of a Conjecture of Euler
cites work: Q3894556
cites work: On 2-Designs Whose Group of Automorphisms is Transitive on the Ordered Pairs of Intersecting Lines
cites work: Q3798841
cites work: Q3759461
cites work: Finite Permutation Groups and Finite Simple Groups
cites work: Q4744888
cites work: 2-transitive and antiflag transitive collineation groups of finite projective spaces
cites work: Inductive subgroups of flag transitive groups
cites work: Groups acting flag-transitively on designs
cites work: 3/2-transitive permutation groups and designs
cites work: A class of Frobenius regular groups
cites work: Block transitive automorphism groups of designs
cites work: Steiner triple systems with block-transitive automorphism groups
cites work: The Trigonometry of GF(2 2n ) and Finite Hyperbolic Planes
cites work: The construction of finite regular hyperbolic planes from inversive planes of even order
cites work: Finite translation planes with collineation groups doubly transitive on the points at infinity
cites work: Flag-transitivity and primitivity
cites work: Q3324542
cites work: Transitivity on ordered pairs of lines in finite linear spaces
cites work: A geometric consequence of the classification of finite doubly transitive groups
cites work: Flag-transitive finite simple groups
cites work: Line-primitive groups of small rank
cites work: Most block-transitive t-designs are point-primitive
cites work: Doubly homogeneous \(2\text{-}(v,k,1)\) designs
cites work: Scharf transitive und scharf fahnentransitive Kollineationsgruppen
cites work: Q5545191
cites work: Some characterizations of finite projective spaces
cites work: The classification of the translation planes of order 16. I
cites work: Q4724811
cites work: Q3699506
cites work: Solvability of groups of odd order
cites work: Q3314470
cites work: Flag-transitive projective planes
cites work: The Flag-Transitive Collineation Groups of the Finite Desarguesian Affine Planes
cites work: Solvable flag transitive affine groups
cites work: Solvable, flag-transitive, rank 3 collineation groups
cites work: Subgroups of prime power index in a simple group
cites work: STEINER TRIPLE SYSTEMS WITH GEOMETRIC MINIMALLY GENERATED SUBSYSTEMS
cites work: Projective Planes
cites work: Steiner triple systems with a doubly transitive automorphism group
cites work: Uniqueness of the Projective Plane of Order Eight
cites work: Eine nicht-desarguesche zweifach transitive affine Ebene der Ordnung 27

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cites work: On finite line transitive affine planes

Finite linear spaces with flag-transitive groups (cites work)
cites work: Transitive linear groups and linear groups which contain irreducible subgroups of prime order

Finite linear spaces with flag-transitive groups (cites work)
cites work: Transitive linear groups and linear groups which contain irreducible subgroups of prime order. II

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cites work: Q3732451

Finite linear spaces with flag-transitive groups (cites work)
cites work: Geometric ABA-groups

Finite linear spaces with flag-transitive groups (cites work)
cites work: Zweifach transitive, auflösbare Permutationsgruppen

Finite linear spaces with flag-transitive groups (cites work)
cites work: On finite affine planes of rank 3

Finite linear spaces with flag-transitive groups (cites work)
cites work: Automorphism groups of designs

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cites work: Line-transitive collineation groups of finite projective spaces

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cites work: Plane geometries associated with certain 2-transitive groups

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cites work: Homogeneous designs and geometric lattices

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cites work: Q3702886

Finite linear spaces with flag-transitive groups (cites work)
cites work: Primitive permutation groups of odd degree, and an application to finite projective planes

Finite linear spaces with flag-transitive groups (cites work)
cites work: Finite affine planes with collineation groups primitive on the points

Finite linear spaces with flag-transitive groups (cites work)
cites work: Note on a consequence for affine groups of the classification theorem for finite simple groups

Finite linear spaces with flag-transitive groups (cites work)
cites work: Steiner triple systems with doubly transitive automorphism groups: A corollary to the classification theorem for finite simple groups

Finite linear spaces with flag-transitive groups (cites work)
cites work: Q3791327

Finite linear spaces with flag-transitive groups (cites work)
cites work: Finite affine planes of rank three are translation planes

Finite linear spaces with flag-transitive groups (cites work)
cites work: A characterization of the Lüneburg planes

Finite linear spaces with flag-transitive groups (cites work)
cites work: Steinersche Tripelsysteme mit fahnentransitiver Kollineationsgruppe

Finite linear spaces with flag-transitive groups (cites work)
cites work: Fahnenhomogene Quadrupelsysteme

Finite linear spaces with flag-transitive groups (cites work)
cites work: Über projektive Ebenen, in denen jede Fahne von einer nicht-trivialen Elation invariant gelassen wird

Finite linear spaces with flag-transitive groups (cites work)
cites work: Scharf fahnentransitive endliche affine Räume

Finite linear spaces with flag-transitive groups (cites work)
cites work: Some remarks concerning the Ree groups of type \((G_2)\)

Finite linear spaces with flag-transitive groups (cites work)
cites work: Q3891315

Finite linear spaces with flag-transitive groups (cites work)
cites work: A Class of Flag Transitive Planes

Finite linear spaces with flag-transitive groups (cites work)
cites work: Q3884734

Finite linear spaces with flag-transitive groups (cites work)
cites work: A New Flag Transitive Affine Plane of Order 27

Finite linear spaces with flag-transitive groups (cites work)
cites work: Q3317664

Finite linear spaces with flag-transitive groups (cites work)
cites work: The translation planes of order twenty-five

Finite linear spaces with flag-transitive groups (cites work)
cites work: Automorphisms of unitary block designs

Finite linear spaces with flag-transitive groups (cites work)
cites work: Q5531588

Finite linear spaces with flag-transitive groups (cites work)
cites work: On projective and affine planes with transitive collineation groups

Finite linear spaces with flag-transitive groups (cites work)
cites work: Fahnentransitive Ebenen gerader Ordnung

Finite linear spaces with flag-transitive groups (cites work)
cites work: Fahnentransitive Ebenen ungerader Ordnung

Finite linear spaces with flag-transitive groups (cites work)
cites work: The Structure of Certain Triple Systems

Finite linear spaces with flag-transitive groups (cites work)
cites work: Flag-Transitive Planes of Even Order

Finite linear spaces with flag-transitive groups (cites work)
cites work: Über Translationsebenen mit Kollineationsgruppen, die die Punkte der ausgezeichneten Geraden zweifach transitiv permutieren

Finite linear spaces with flag-transitive groups (cites work)
cites work: On a Method of Construction of Partial Geometries and Partial Bolyai- Lobachevsky Planes

Finite linear spaces with flag-transitive groups (cites work)
cites work: Unitary block designs

Finite linear spaces with flag-transitive groups (cites work)
cites work: Ovoides et groupes de Suzuki

Finite linear spaces with flag-transitive groups (cites work)
cites work: On perspectivities of finite projective planes

Finite linear spaces with flag-transitive groups (cites work)
cites work: On finite affine line transitive planes

Finite linear spaces with flag-transitive groups (cites work)
cites work: Fahnentransitive Automorphismengruppen von Blockplänen. (Flag transitive automorphism groups of block designs.)

Finite linear spaces with flag-transitive groups (cites work)
cites work: Das kleinste Ree-Unital. (The smallest Ree-unital)

Finite linear spaces with flag-transitive groups (cites work)
cites work: Q4405355

Finite linear spaces with flag-transitive groups (cites work)
cites work: Spreads, Translation Planes and Kerdock Sets. II

Finite linear spaces with flag-transitive groups (cites work)
cites work: An exponential number of generalized Kerdock codes

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