A seven-term exact sequence for the cohomology of a group extension

TL;DR

This paper constructs a complete seven-term exact sequence for group extension cohomology using elementary methods, clarifying the maps involved and enhancing understanding of the sequence's structure.

Contribution

It provides an elementary construction of the seven-term exact sequence, explicitly describing all maps and interpretations, improving usability over spectral sequence approaches.

Findings

Explicit description of all maps in the sequence

Elementary construction using conjugation in groups

Enhanced understanding of cohomology interpretations

Abstract

In this paper, we construct a seven-term exact sequence involving the cohomology groups of a group extension. Although the existence of such a sequence can be derived using spectral sequence arguments, there is little knowledge about some of the maps occuring in the sequence, limiting its usefulness. Here we present a construction using only very elementary tools, always related to the notion of conjugation in a group. This results in a complete and usable description of all the maps, which we describe both on cocycle level as on the level of the interpretations of low dimensional cohomology groups (e.g. group extensions).