K_1 of a p-adic group ring I. The determinantal image

TL;DR

This paper investigates the structure of the K_1 group of a p-adic group ring, focusing on the determinantal image and employing a Frobenius lift to establish a fixed point theorem.

Contribution

It introduces a novel approach using the group logarithm with Frobenius lift to analyze the determinantal image of K_1 in p-adic group rings.

Findings

Proves a fixed point theorem for the determinantal image of K_1.

Defines the group logarithm using Frobenius lift in this context.

Provides new insights into the structure of K_1 of p-adic group rings.

Abstract

We study the K-group K_1 of the group ring of a finite group over a coefficient ring which is p-adically complete and admits a lift of Frobenius. In this paper, we consider the image of K_1 under the determinant map; the central tool is the group logarithm which we can define using the Frobenius lift. Using this we prove a fixed point theorem for the determinantal image of K_1.