Iterated group extensions

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.