The Group Logarithm Past and Present

Abstract

We give an account of the group logarithm and discuss a number of its applications.We begin by rehearsing some basic theory, and then go on to consider its use both in the theory of class groups of group rings and in the Galois structure of rings of algebraic integers.We conclude with a brief insight into its recent use in the construction of an adelic second Chern class for a non-commutative Riemann Roch theorem. The use of the group logarithm in non-commutative Iwasawa theory is discussed elsewherein this volume. © Springer-Verlag Berlin Heidelberg 2013.