The mod-2 cohomology of metacyclic 2-groups
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.
mod-2 cohomology rings of finite 2-groups
scientific article; zbMATH DE number 4011427