Iterated group extensions

TL;DR

This paper introduces the concept of iterated group extensions, interprets them via group cohomology, and explicitly identifies cohomology groups and morphisms in the associated long exact sequence.

Contribution

It defines iterated group extensions and provides a cohomological framework with explicit identifications, linking extensions, automorphisms, and outer actions.

Findings

Explicit descriptions of cohomology groups in iterated extensions

Identification of morphisms in the long exact sequence

Relations between extensions, automorphisms, and outer actions

Abstract

We introduce the notion of iterated group extensions, which, roughly speaking, is what one obtains by forming a group extension of a group extension. We interpret iterated extensions in terms of group cohomology, in the same way as Eilenberg-MacLane did for usual group extensions. From the E_2-spectral sequence of a group extension, there is a 6-term long exact sequence in which various cohomology groups of degree 1 or 2 appear. We give an explicit identification of each cohomology group and each morphism appearing in this long exact sequence in terms of iterated extensions and associated notions. These identifications enable us to uncover natural relations between (iterated) extensions, their automorphism groups, and their outer actions.