On the cohomology and homotopy of Swan groups
type: Item
instance of: scholarly article
title: On the cohomology and homotopy of Swan groups
zbMATH Open document ID: 0882.20034
DOI: 10.1007/PL00004312
author: John Martino
author: Stewart Priddy
published in: Mathematische Zeitschrift
publication date: 1998-03-10T00:00:00Z
review text: If \(P\) is a Sylow \(p\)-subgroup of a finite group \(G\), then there is an injective map \(H^*(G;\mathbb{F}_p) @>\text{res}>> H^*(P;\mathbb{F}_p)^{W_G(P)}\), where \(W_G(P)=N_G(P)/PC_G(P)\) and cohomology is taken with simple coefficients in \(\mathbb{F}_p\). A Swan group is a finite \(p\)-group \(P\) for which this map is always an isomorphism, for any finite group \(G\) with \(P\) as Sylow \(p\)-subgroup. The authors establish a sufficient condition for a \(p\)-group to be a Swan group and show that for \(p\) odd, every metacyclic \(p\)-group is a Swan group. They then show that if the classifying space \(BP\) has no stable summands involving the transfer, then \(P\) is a Swan group. Finally, using U. Martin's definition of ``almost all'', they show that almost all \(p\)-groups are Swan groups.
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 1033934
zbMATH Keywords: Sylow \(p\)-subgroups
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: metacyclic \(p\)-groups
zbMATH Keywords: finite groups
zbMATH Keywords: cohomology
zbMATH Keywords: Swan groups
zbMATH Keywords: classifying spaces
zbMATH Keywords: transfer
reviewed by: T. W. Hungerford
MaRDI profile type: Publication
full work available at URL: pl00004312
OpenAlex ID: W2019572716
description: scientific article; zbMATH DE number 1033934