Restrictions on the structure of subgroup lattices of finite alternating and symmetric groups.
type: Item
instance of: scholarly article
title: Restrictions on the structure of subgroup lattices of finite alternating and symmetric groups.
zbMATH Open document ID: 1185.20020
author: Michael Aschbacher
author: John Shareshian
published in: Journal of Algebra
publication date: 2010-01-08T00:00:00Z
review text: The paper deals with an important and well-known open question: which finite lattices can be represented as intervals in the subgroup lattice of a finite group? In particular the authors exhibit restrictions on the structure of the lattice \(\mathcal O_G(H)\) of subgroups of \(G\) containing \(H\leq G\) when \(G\) is a finite alternating or symmetric group. For a finite lattice \(L\), let \(L'\) be the poset obtained by removing the minimum and maximum elements of \(L\) and, for \(n\in\mathbb{N}\), let \(\Delta(n)\) be the lattice of all subsets of \(\{1,\dots,n\}\), ordered by inclusion. For positive integers \(m_1,\dots,m_t\), the authors consider the lattice \(D\Delta(m_1,\dots,m_t)\) defined as follows: \(D\Delta(m_1,\dots,m_t)'\) is a disjoint union of posets \(\mathcal C_1,\dots,\mathcal C_t\) with \(C_i\cong\Delta(m_i)'\) for all \(i\) and if \(i\neq j\) then no element of \(C_i\) is comparable to any element of \(C_j\). The authors prove that if \(t>1\) and \(m_1\geq m_2\geq\cdots\geq m_t\geq 3\), then \(G\in\{\text{Alt}(n),\text{Sym}(n)\}\) contains no subgroup \(H\) with \(\mathcal O_G(H)\cong D\Delta(m_1,\dots,m_t)\).
reviewed by: Andrea Lucchini
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20B35
Mathematics Subject Classification ID: 06B15
Mathematics Subject Classification ID: 08A30
zbMATH DE Number: 5655017
zbMATH Keywords: subgroup lattices
zbMATH Keywords: symmetric groups
zbMATH Keywords: alternating groups
zbMATH Keywords: intervals
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2009.05.042
OpenAlex ID: W2008673225
cites work: Overgroups of primitive groups. II.
cites work: OVERGROUPS OF PRIMITIVE GROUPS
cites work: On intervals in subgroup lattices of finite groups
cites work: Signalizer lattices in finite groups.
cites work: The classification of the finite simple groups
cites work: Maximal subgroups of finite groups
cites work: On representing finite lattices as intervals in subgroup lattices of finite groups
cites work: Q4882944
cites work: Subgroups of prime power index in a simple group
cites work: On the minimal degree of a primitive permutation group
cites work: Q4114031
cites work: A classification of the maximal subgroups of the finite alternating and symmetric groups
cites work: Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups
cites work: Topology of order complexes of intervals in subgroup lattices.
cites work: On the Möbius number of the subgroup lattice of the symmetric group
DOI: 10.1016/J.JALGEBRA.2009.05.042
description: scientific article; zbMATH DE number 5655017