The mod-2 cohomology of metacyclic 2-groups
type: Item
instance of: scholarly article
title: The mod-2 cohomology of metacyclic 2-groups
zbMATH Open document ID: 0623.55006
DOI: 10.1016/0022-4049(87)90035-1
author: David J. Rusin
published in: Journal of Pure and Applied Algebra
publication date: 1987-01-01T00:00:00Z
review text: We calculate the mod-2 cohomology rings \(H^*(G\), \({\mathbb{F}}_ 2)\) of finite 2-groups G which are cyclic extensions of cyclic groups. We view these groups as extensions of normal 4-groups and apply the Eilenberg- Moore spectral sequence, which we describe as it applies to group cohomology. Unlike the Serre spectral sequence, this is always observed to collapse, and the extension problems to be solved are minor. Moreover, it enables us to deduce the structure of the cohomology rings as modules over the Steenrod algebra. (In subsequent papers we will apply this techniques to other groups as well.) The results of the calculation are themselves not remarkable; they resemble the results obtained for special cases by Diethelm. Most unusual are the groups with cyclic subgroups of index 2, whose cohomology rings are well known; the present paper has the advantage of explaining why these groups behave exceptionally. Huebschmann has since used a different technique to generalize the calculations to metacyclic groups other than 2-groups; however, his approach is perhaps more limited to metacyclic groups than ours is.
Mathematics Subject Classification ID: 55R40
Mathematics Subject Classification ID: 55T20
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 55S10
zbMATH DE Number: 4011427
zbMATH Keywords: mod-2 cohomology rings of finite 2-groups
zbMATH Keywords: Eilenberg-Moore spectral sequence
zbMATH Keywords: Steenrod algebra
MaRDI profile type: Publication
full work available at URL: 0022-4049(87)90035-1
OpenAlex ID: W2045115910
cites work: Q3255397
cites work: Generalizations of Certain Elementary Theorems on p -Groups
cites work: The mod p cohomology rings of the nonabelian split metacyclic p-groups
cites work: On the Künneth theorem. I: The Eilenberg-Moore spectral sequence
description: scientific article; zbMATH DE number 4011427