Topology of subgroup lattices of symmetric and alternating groups.
type: Item
instance of: scholarly article
title: Topology of subgroup lattices of symmetric and alternating groups.
zbMATH Open document ID: 1040.20019
DOI: 10.1016/S0097-3165(03)00138-9
author: John Shareshian
published in: Journal of Combinatorial Theory. Series A
publication date: 2004-01-06T00:00:00Z
review text: When \(G\) is a non-solvable finite group its order complex \(\Delta(G)\) is mostly an unknown entity other than what certain forays, as executed in this interesting paper, have managed to demonstrate. By looking at \(\Delta(S_n)\) and \(\Delta(A_n)\) when \(n\) is prime or for \(\Delta(S_n)\) when \(n\) is a power of 2, more detailed information about these complexes shows them as wedges of spheres of (not always the same) certain dimensions particularizing the information in the \(\Delta(S_n)\) case for \(n>1\) contained in Theorem 1.1, with the partition lattice playing an important role and noting that \(\dim(\widetilde H_{n-3} (\Delta(S_n)))\geq\tfrac{n!}2\) for all \(n>1\) as a consequence of Theorem 1.1 as well. Conjecture 1.3 is then also one of natural expectation as well as curiosity which points ahead at directions to follow using methods and techniques which have become if not all standard at least recognizable when first encountered by the somewhat cognoscenti in this area.
reviewed by: J. Neggers
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 06B30
Mathematics Subject Classification ID: 05E25
Mathematics Subject Classification ID: 55P15
Mathematics Subject Classification ID: 20J05
zbMATH DE Number: 2022048
zbMATH Keywords: subgroup lattices
zbMATH Keywords: posets
zbMATH Keywords: order complexes
zbMATH Keywords: homotopy types
zbMATH Keywords: homology
zbMATH Keywords: symmetric groups
zbMATH Keywords: alternating groups
zbMATH Keywords: partition lattices
MaRDI profile type: Publication
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full work available at URL: s0097-3165(03)00138-9
OpenAlex ID: W2171488644
description: scientific article; zbMATH DE number 2022048