On primitive overgroups of quasiprimitive permutation groups
type: Item
instance of: scholarly article
title: On primitive overgroups of quasiprimitive permutation groups
zbMATH Open document ID: 1024.20002
DOI: 10.1016/S0021-8693(03)00113-3
author: Robert W. Baddeley
author: Cheryl E. Praeger
published in: Journal of Algebra
publication date: 2003-06-18T00:00:00Z
review text: A permutation group is called quasiprimitive if all its non-trivial normal subgroups are transitive. It is elementary that any primitive group is quasiprimitive. The authors investigate the pairs \((G,H)\) of permutation groups of finite degree such that \(G\leq H\) with \(G\) quasiprimitive and \(H\) primitive. Since the second author already studied the case where \(G\) is primitive [in \textit{C. E. Praeger}, Proc. Lond. Math. Soc., III. Ser. 60, No. 1, 68--88 (1990; Zbl 0653.20005)], they assume in this paper \(G\) imprimitive. They partition the class of quasiprimitive groups into 8 families, 3 of which containing only primitive groups, corresponding to 8 families of primitive groups. Hence there are 40 possible types of embeddings of \(G\) imprimitive quasiprimitive into \(H\) primitive. A lot of these cases are impossible or exist only if \(G\) is primitive. The existing embeddings are described, except in the cases where \(H\) is either almost simple or the blow-up of an almost simple group. The results rely on the classification of the finite simple groups.
reviewed by: Alice Devillers
Mathematics Subject Classification ID: 20B15
zbMATH DE Number: 1930294
zbMATH Keywords: permutation groups
zbMATH Keywords: primitive groups
zbMATH Keywords: quasiprimitive groups
zbMATH Keywords: embeddings
MaRDI profile type: Publication
cites work: Primitive Permutation Groups with a Regular Non-Abelian Normal Subgroup
cites work: Two-arc transitive graphs and twisted wreath products
cites work: On classifying all full factorisations and multiple-factorisations of the finite almost simple groups
cites work: Intransitive Cartesian decompositions preserved by innately transitive permutation groups
cites work: Finite Permutation Groups and Finite Simple Groups
cites work: Line-transitive, point quasiprimitive automorphism groups of finite linear spaces are affine or almost simple.
cites work: Q3684278
cites work: Finite two-are transitive graphs admitting a ree simple group
cites work: Fintte two-arc transitive graphs admitting a suzuki simple group
cites work: Linear Groups with Orders Having Certain Large Prime Divisors
cites work: Subgroups of prime power index in a simple group
cites work: Fixed Points of Automorphisms of Certain Locally Finite Groups and Chevalley Groups
cites work: Two-arc transitive graphs admitting a two-dimensional projective linear group
cites work: On finite affine 2-arc transitive graphs
cites work: Q3952291
cites work: \(k\)-homogeneous groups
cites work: Q3996618
cites work: On the point stabilizer in a primitive permutation group.
cites work: Primitive Subgroups of Wreath Products in Product Action
cites work: The finite vertex-primitive and vertex-biprimitive \(s\)-transitive graphs for \(s\geq 4\)
cites work: On finite \(s\)-transitive graphs of odd order
cites work: The Affine Permutation Groups of Rank Three
cites work: A classification of the maximal subgroups of the finite alternating and symmetric groups
cites work: Q3791327
cites work: The maximal factorizations of the finite simple groups and their automorphism groups
cites work: On factorizations of almost simple groups
cites work: Transitive subgroups of primitive permutation groups
cites work: Twisted wreath products of groups
cites work: Q4705904
cites work: The Inclusion Problem for Finite Primitive Permutation Groups
cites work: An O'Nan-Scott Theorem for Finite Quasiprimitive Permutation Groups and an Application to 2-Arc Transitive Graphs
cites work: Q3907748
Wikidata QID: Q56987906
full work available at URL: s0021-8693(03)00113-3
OpenAlex ID: W2021446986
description: scientific article; zbMATH DE number 1930294