On invariant (co)homology of a group

TL;DR

This paper explores various notions of (co)homology for groups with automorphism actions, examining their relations to semidirect product homology, abelianization, and extension classification.

Contribution

It clarifies the connections between invariant (co)homology notions and classical group (co)homology, providing new insights into group extensions and abelianization.

Findings

Relations between invariant (co)homology and semidirect product (co)homology

Interpretation of first homology as a form of abelianization

Classification results for invariant group extensions

Abstract

There are different notions of homology and cohomology that can be defined for a group with an action of another group by group automorphisms. In this paper we address three natural questions that arise in this context. Namely, the relation of these notions with the usual (co)homology of a semidirect product, the interpretation of the first homology group as some kind of abelianization and the classification of (invariant) group extensions.