On Quillen's conjecture for the \(p\)-groups complex
type: Item
instance of: scholarly article
title: On Quillen's conjecture for the \(p\)-groups complex
zbMATH Open document ID: 0782.20039
DOI: 10.2307/2946530
author: Michael Aschbacher
author: Stephen D. Smith
published in: Annals of Mathematics. Second Series
publication date: 1994-03-10T00:00:00Z
review text: If \(G\) is a finite group and \(p\) a prime, then \({\mathcal S}_ p(G)\) is the simplicial complex consisting of all nontrivial \(p\)-subgroups of \(G\). \textit{D. Quillen} [Adv. Math. 28, 101-128 (1978; Zbl 0388.55007)] proved that \({\mathcal S}_ p(G)\) is homotopy equivalent to its subcomplex \({\mathcal A}_ p(G)\) consisting of the nontrivial elementary \(p\)-subgroups and that \({\mathcal A}_ p(G)\) is contractible provided \(O_ p(G)\neq 1\). He conjectured also that the converse statement should hold: if \(O_ p(G) = 1\), then \(\widetilde{\text{H}}_*({\mathcal A}_ p(G))\neq 0\). The main result of the paper under review is the following Theorem: Quillen's conjecture holds when \(p>5\) and \(G\) has no components \(U_ n(q)\) with \(q\equiv-1\pmod p\) and \(p\) odd.
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 55P10
Mathematics Subject Classification ID: 20D30
zbMATH DE Number: 205463
zbMATH Keywords: finite group
zbMATH Keywords: simplicial complex
zbMATH Keywords: \(p\)-subgroups
zbMATH Keywords: homotopy equivalent
zbMATH Keywords: contractible
zbMATH Keywords: Quillen's conjecture
reviewed by: S. A. Syskin
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2946530
OpenAlex ID: W2312969627
Wikidata QID: Q123146185
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description: scientific article; zbMATH DE number 205463