Overgroups of primitive groups. II.
type: Item
instance of: scholarly article
title: Overgroups of primitive groups. II.
zbMATH Open document ID: 1190.20013
author: Michael Aschbacher
published in: Journal of Algebra
publication date: 2009-10-13T00:00:00Z
review text: Assume that \(\Omega\) is a set of finite order \(n\), \(G\in\{\text{Sym}(\Omega),\text{Alt}(\Omega)\}\), and \(H\) is a subgroup of \(G\) acting primitively on \(\Omega\). In this paper it is continued the study of the set \(\mathcal O_G(H)\) of overgroups of \(H\) in \(G\) [for part I cf. J. Aust. Math. Soc. 87, No. 1, 37-82 (2009; Zbl 1190.20012)]. Let \(\Lambda\) be a finite lattice. Then \(\Lambda\) has a greatest member \(\infty\) and a least member \(0\). Set \(\Lambda'=\Lambda-\{0,\infty\}\). Regard \(\Lambda\) as a graph with adjacency relation the comparability relation, and define \(\Lambda\) to be disconnected if the graph \(\Lambda'\) is disconnected. Write \(\Delta(m)\) for the lattice of subsets of an \(m\)-set, partially ordered by inclusion. Define a \(D\Delta\)-lattice to be a lattice \(\Lambda\) such that \(\Lambda'\) has \(r>1\) connected components \(\Lambda_i'\simeq\Delta(m_i)\) for some \(m_i>2\). Aschbacher and Shareshian conjecture that no \(D\Delta\)-lattice is an interval in the subgroup lattice of a finite group and Aschbacher reduced the verification of this conjecture to two problems about the overgroup lattices in almost simple groups. The first problem is to prove that if \(X\) is an almost simple group and \(Y\leq X\) then \(\mathcal O_X(Y)\) is not a \(D\Delta\)-lattice. This is accomplished, when \(X\) is an alternating or symmetric group and \(Y\) is primitive (Theorem E).
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20B35
Mathematics Subject Classification ID: 20B15
Mathematics Subject Classification ID: 06B15
Mathematics Subject Classification ID: 20E28
zbMATH DE Number: 5614722
zbMATH Keywords: primitive permutation groups
zbMATH Keywords: lattices of subgroups
zbMATH Keywords: maximal subgroups
zbMATH Keywords: finite lattices
zbMATH Keywords: finite alternating groups
zbMATH Keywords: finite symmetric groups
reviewed by: Alexandre A. Makhnev
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2009.04.044
OpenAlex ID: W2057163445
cites work: On the maximal subgroups of the finite classical groups
cites work: On intervals in subgroup lattices of finite groups
cites work: Signalizer lattices in finite groups.
cites work: OVERGROUPS OF PRIMITIVE GROUPS
cites work: Overgroups of Sylow subgroups in sporadic groups
cites work: A characterization of some finite groups of characteristic 3
cites work: Q4833271
cites work: On representing finite lattices as intervals in subgroup lattices of finite groups
cites work: Q3708174
cites work: Chevalley groups of type \(G_ 2\) as the group of a trilinear form
cites work: Q4243089
cites work: The maximal factorizations of the finite simple groups and their automorphism groups
cites work: A classification of the maximal subgroups of the finite alternating and symmetric groups
cites work: On the Overgroups of Irreducible Subgroups of the Finite Classical Groups
cites work: Some groups generated by transvections
cites work: Some subgroups of \(SI_ n(F_2)\)
cites work: The Inclusion Problem for Finite Primitive Permutation Groups
cites work: Intervals in the subgroup lattices of the alternating and symmetric groups of prime degree
cites work: Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups
cites work: Q4295765
cites work: Q3949002
cites work: Determination of the finite primitive reflection groups over an arbitrary field of characteristic not 2. III
DOI: 10.1016/J.JALGEBRA.2009.04.044
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description: scientific article; zbMATH DE number 5614722