Cohomology of Metacyclic Groups

Abstract

Abstract. Let e: 1 -> N -► G -► K -> 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 Z over the group ring ZG 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 e. The calculation is then completed by comparing the two bounds.

The Lie Algebra Perturbation Lemma

“…Suitable HPT constructions that are compatible with other algebraic structure enabled us to carry out complete numerical calculations in group cohomology [10], [11], [12] which cannot be done by other methods.…”

Homological models for semidirect products of finitely generated Abelian groups

“…The homological perturbation theory has been applied to compute resolutions for a wide range of groups (e.g. finitely generated two-step nilpotent groups [24], metacyclic groups [25], finite p-groups [18]). Using homological perturbation theory, we show that a resolution R of Z Z over Z Z[K × χ H ] (which splits off of the bar resolution) arises from a homological model for K × χ H .…”

A Mathematica Notebook for Computing the Homology of Iterated Products of Groups

“…Furthermore, our method may be extended to cover iterated products of other groups for which homological models are known, such as finitely generated torsion-free nilpotent groups [17,15], finite p-groups [16,10], finitely generated two-step nilpotent groups [12] and metacyclic groups [13].…”