The mod-p cohomology rings of metacyclic groups
type: Item
instance of: scholarly article
title: The mod-p cohomology rings of metacyclic groups
zbMATH Open document ID: 0688.20032
DOI: 10.1016/0022-4049(89)90107-2
author: J. Huebschmann
published in: Journal of Pure and Applied Algebra
publication date: 1989-01-01T00:00:00Z
review text: In this very interesting paper the author obtains a complete picture for the mod \(p\) cohomology ring of any finite metacyclic group. If \(\varepsilon: 1\to N\to G\to K\to 1\) is a group extension and you are given free resolutions for \(N\) and \(K\) then \textit{C. T. C. Wall} [in Proc. Camb. Philos. Soc. 57, 251-255 (1961; Zbl 0106.249)] showed how to construct from them a free resolution for \(G\), which may also be used to give the spectral sequence of the extension \(\varepsilon\). Here the author elaborates on Wall's free resolution and by using homological perturbation theory he does give explicit construction for the differential in low dimensions, in the case where \(K\) and \(N\) are finite cyclic groups. He then uses this free resolution to compute the mod \(p\) cohomology ring of \(G\) via the spectral sequence of the extension \(\varepsilon\). There are many interesting results in this paper, but their statements and proofs are too technical to elaborate on in this review.
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 20E22
zbMATH DE Number: 4127509
zbMATH Keywords: mod p cohomology rings
zbMATH Keywords: finite metacyclic groups
zbMATH Keywords: group extensions
zbMATH Keywords: free resolutions
zbMATH Keywords: spectral sequences
zbMATH Keywords: perturbations
zbMATH Keywords: differentials
reviewed by: Olympia Talelli
MaRDI profile type: Publication
full work available at URL: 0022-4049(89)90107-2
OpenAlex ID: W2081975357
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Wikidata QID: Q127014983
description: scientific article; zbMATH DE number 4127509