Group Extensions, Representations, and the Schur Multiplicator

The aim of these notes is a unified treatment of various grouptheoretic topics for which, as it turns out, the Schur multiplicator is the key. At the beginning of this century, classical projective geometry was at its peak, while representation theory was growing in the hands of Frobenius and Burnside. In this climate our subject started with the two important papers of Schur ~I~,~2~ on the projective representations of finite groups. But it was only in the light of the much more recent (co)homology theory of groups that the true nature of Schur's "Multlplicator" and its impact on group theory was fully realized; the papers by GREEN EI~, YAMAZAKI EI~, STALLINGS ~I~, and STAMMBACH ~I~ have been most influential in this regard.
The first chapter provides the setting for these notes. We start out with the concepts of group extension (handled in terms of diagrams) and Schur multiplicator (here defined by the Schur-Hopf Formula) to obtain a group-theoretlc version of the Universal Coefficient Theorem. All these concepts and the Ganea map have a homological flavor, but are here developed in a rather elementary group-theoretic fashion; the (co)homology theory of groups is not a prerequisite for reading most of these notes. The first chapter also includes a full translation from our approach to the usual group (co)homology for the reader's convenience. (We feel that our presentation is very suited for the applications to follow, but this view is to some extent a matter of taste.)