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arXiv · 1903.03080

The conjugation action in completed group rings

William Woods
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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 $\Gamma$.

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William Woods. 2019-03-07. The conjugation action in completed group rings. https://arxiv.org/abs/1903.03080

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