6 - THE GROUP LOGARITHM

Summary

In the Hom–descriptions of Grothendieck groups of group rings given in chapter 1, we obtain factors of the type Det(ℤ Г) in the denominator. Thus, in order to be able to see how close a representing homomorphism lies to the denominator, we have to have a much better understanding of Det(ℤ Г^*). This is achieved by means of the group logarithm.

In the first section we begin by stating the main results concerning the group logarithm, for the case of p–groups. We then prove a number of results, but we defer the proof of the integrality property of the logarithm until section 2. In section 3 we extend our techniques to deal with Q– p–elementary groups. The relevance of this tool for classgroups of group rings will become more clear in subsequent chapters, where it is used to prove several major results.

Nearly all the results of this chapter are taken from [T1] – though it should be mentioned that the origins of this work are to be found in [T2].

THE MAIN RESULTS

Let Г be a finite p–group, let K be a finite non-ramified extension of Q_p, and let Δ = Gal(K/Q_p).