# The conjugation action in completed group rings

## TL;DR

This paper investigates the conjugation action in completed group rings of p-valuable groups, establishing structural links between automorphisms of the algebra and the underlying group, and calculating fixed rings under subgroup actions.

## Contribution

It proves that automorphisms of the completed group algebra induced by conjugation correspond to automorphisms of the group itself and computes fixed rings for subgroup conjugation actions.

## Key findings

- Inner automorphisms of $kG$ correspond to those of $G$.
- Calculated $ ext{Gamma}$-fixed rings for certain ideals.
- Established structural links between algebra automorphisms and group automorphisms.

## Abstract

Let $k = \mathbb{F}_p$ or $\mathbb{Z}_p$ (or finite extensions of these). Let $G$ be a $p$-valuable group, and form its completed group algebra $kG$. By analysing the conjugation action of $G$ on itself, we prove two structural results. Firstly, we show that all inner automorphisms of $kG$ that preserve $G$ are induced from inner automorphisms of $G$. Secondly, for a closed subgroup $\Gamma$ of $G$, we calculate the $\Gamma$-fixed ring of $kG/I$ under the conjugation action of $\Gamma$, for certain ideals $I$ induced from the $G$-centraliser of $\Gamma$.