2 - Applications of cohomology to the theory of groups

Summary

INTRODUCTION

Homological concepts have long been used implicitly in the theory of groups. They occur, for example, in the work of Hölder (1895) and Schreier (1926) on group extensions and of Schur (1904, 1907) on projective representations. The significance for group theory of the cohomology groups in low dimensions appears to have first been recognized by Eilenberg and MacLane in the 1940's. On the other hand, actual applications of homology to establish theorems of a purely group theoretical nature are of more recent origin, dating largely from the last twenty five years. Perhaps the most famous result to be discovered by the use of such methods is Gaschütz's theorem on the existence of outer automorphisms of finite p-groups (1965-6).

Recently there has been an increasing awareness among group theorists of the utility of homology, and many have been led to equip themselves with homological tools. In this respect the lecture notes of K. W. Gruenberg and U. Stammbach have been influential.

The present work is not intended to be a survey; rather its aim is to review the ways in which the cohomology groups arise in group theory and then to exploit this connection by proving some theorems about groups.