Automorphisms and cohomology

TL;DR

This paper generalizes an existing exact sequence relating automorphisms and cohomology from centric to arbitrary group extensions, providing new insights into automorphism groups and their properties.

Contribution

It extends the known exact sequence for automorphisms and cohomology from centric extensions to all group extensions, revealing broader applicability and new consequences.

Findings

Derived a generalized exact sequence for automorphisms of arbitrary extensions.

Connected the generalized sequence to previous results for centric extensions.

Provided criteria for the solvability of automorphism groups.

Abstract

Let 1-> H -> G _> Q -> 1 be an exact sequence of groups. In the paper of R. Oliver and J. Ventura, TAMS,362(2009), the following exact sequence was developed for centric extensions, i.e the centralizer of H in G is contained in H, 0-> H^1(Q,zH) -> Aut(G,H) -> N_{Out H}(F Q)/F Q -> H^2(Q,zH) where Aut(G,H) are the automorphisms of G which restrict to an automorphism of H, F:Q -> Out H is the outer action determined by the extension, zH is the center of H with Q-action coming from F and N_{Out H} the normalizer. It is the aim of this paper to generalize the above sequence to arbitrary extensions, show how the above result is derived from the general exact sequence and derive other consequences of the general result including determining solvability of Aut(G,H).