Abstract

We give a historical perspective on the Wells exact sequence and Buckley's interpretation of it, and include a survey of applications and extensions of their work from the 1970's to the present.

Introduction

In 1971, CharlesWells [26] constructed an exact sequence for the automorphism group of a group extension. The paper received a bit of attention at the time, but seems to have been largely ignored until the last decade. In particular, a series of papers in the Journal of Algebra over the last few years (see [12], [13], [16]) have brought attention to the sequence and its applications.

This paper presents a historical view of the Wells exact sequence, emphasizing the game-changing nature of Joseph Buckley's interpretation of Wells’ result in the context of group actions. Indeed, one goal of this paper is to give Buckley credit–at least equal to Wells–for describing an invaluable method for investigating automorphisms of group extensions. A survey of applications of Wells’ and Buckley's work is provided.

The paper is organized as follows: Section 2 begins with background information on group extensions and their automorphisms; Section 3 focuses on the 1970's and gives a history of Wells’ work and subsequent papers by Buckley and others; Section 4 focuses on the “resurrection” of Wells’ and Buckley's work in the last decade, but does not contain detailed statements of theorems; and Section 5 concludes with a survey of applications and extensions of the Wells exact sequence, including details missing in Section 4.

Background Information

In this section we give background information on group extensions and their automorphisms. This information is well known and available in many other places, but we will use this opportunity to establish some notation. For group extensions, we will follow the notation in [20] as closely as possible since Derek Robinson's book is widely available, and because both the book and an old paper of his [18] are commonly used resources by those in the field.