K_1 of some noncommutative group rings

TL;DR

This paper extends previous calculations of K_1 groups of p-adic group rings for non-abelian p-groups using abelian subquotients, with implications for non-commutative Iwasawa theory.

Contribution

It generalizes earlier work on K_1 of p-adic group rings to broader classes of non-abelian groups using subgroup and quotient structures.

Findings

Computed K_1 groups up to p-power torsion for non-abelian p-groups

Expressed K_1 in terms of p-adic group rings of abelian subquotients

Provides tools for applications in non-commutative Iwasawa theory

Abstract

In this article I generalise previous computations (by K. Kato, T. Hara and myself) of K_1 (only up to p-power torsion) of p-adic group rings of finite non-abelian p-groups in terms of p-adic group rings of abelian subquotients of the group. Such computation have applications in non-commutative Iwasawa theory due to a strategy proposed by D. Burns, K. Kato (and a modification by T. Hara) for deducing non-commutative main conjectures from commutative main conjectures and certain congruences between special L-values. However, I do not say anything about Iwasawa theory in this article. The details of applications in Iwasawa theory will be presented in a separate paper.