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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Cohomology of metacyclic groups

Author: Johannes Huebschmann
Journal: Trans. Amer. Math. Soc. 328 (1991), 1-72
MSC: Primary 20J06
MathSciNet review: 1031239
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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.

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Keywords: Cohomology of group extensions, cohomology of finite groups, homological perturbation theory
Article copyright: © Copyright 1991 American Mathematical Society