Cohomology of Metacyclic Groups

Abstract

Let ${\mathbf {e}}:1 \to N \to G \to K \to 1$ be an extension of a finite cyclic group $N$ by a finite cyclic group $K$. Using homological perturbation theory, we introduce the beginning of a free resolution of the integers ${\mathbf {Z}}$ over the group ring ${\mathbf {Z}}G$ of $G$ in such a way that the resolution reflects the structure of $G$ as an extension of $N$ by $K$, and we use this resolution to compute the additive structure of the integral cohomology of $G$ in many cases. We proceed by first establishing a number of special cases, thereafter constructing suitable cohomology classes thereby obtaining a lower bound, then computing characteristic classes introduced in an earlier paper, and, finally, exploiting these classes, obtaining upper bounds for the cohomology via the integral cohomology spectral sequence of the extension ${\mathbf {e}}$. The calculation is then completed by comparing the two bounds.