Basic constructions in group extension theory

Abstract

This chapter reviews basic constructions in group extension theory. In many categories of groups with additional structure, the appropriate adaptation of cohomology theory is subject to severe limitations and involves considerable technical difficulties, while the adaptation of much group extension theory is easy. To secure the validity of the results in various important categories of groups, one must take special care to free all the definitions and proofs from eventually inadmissible operations involving group algebras, cross sections, or factor sets. Because of this, a certain number of basic constructions become dominant, and the results express exactness properties of these constructions, most of which remain significant also for group extensions with non-abelian kernels. This consists of a category of groups built over an auxiliary category of spaces, subject to suitable assumptions. It is designed to comprise, at least, the most important categories of groups admitting a strong group extension theory, such as locally compact separable topological groups, affine or general algebraic groups over an algebraically closed field of characteristic 0, and Lie groups with countable component groups.