The mod-p cohomology rings of metacyclic groups
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.
scientific article; zbMATH DE number 4127509
scientific article