Homology of certain sets of 2-subgroups of symmetric groups
type: Item
instance of: scholarly article
title: Homology of certain sets of 2-subgroups of symmetric groups
zbMATH Open document ID: 0786.55005
DOI: 10.1016/S0021-8693(05)80054-7
published in: Journal of Algebra
publication date: 1992-09-27T00:00:00Z
review text: In his paper [Adv. Math. 28, 101-128 (1978; Zbl 0388.55007)], \textit{D. Quillen} showed that the set of non-trivial \(p\)-subgroups of a finite group, \(s_ p(G)\), has the same homotopy type as the set of non-trivial elementary abelian \(p\)-subgroups. In this paper the author initiates a study of the group \(d_ 2(n)\) of non-trivial elmentary abelian 2- subgroups of the symmetric group \(S_ n\), and relates it to \(s_ 2(S_ n)\). Much of the remainder of the paper is devoted to homology calculations. For example it is found that certain of these groups are cyclic of order 3. The article concludes with the observation that although it is difficult to compute the homology of \(d_ 2(n)\) exactly, it is easy to find its Lefschetz module.
Mathematics Subject Classification ID: 55P10
Mathematics Subject Classification ID: 20J99
zbMATH DE Number: 63899
zbMATH Keywords: elmentary abelian 2-subgroups
zbMATH Keywords: symmetric group
zbMATH Keywords: homology
author: Serge Bouc
reviewed by: Solomon Jekel
MaRDI profile type: Publication
cites work: Q3681031
cites work: Exponentiation and Steinberg modules
cites work: The representation theory of the symmetric groups
cites work: Homotopy properties of the poset of nontrivial p-subgroups of a group
description: scientific article; zbMATH DE number 63899