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Subgroup rigidity in finite-dimensional group algebras over $ p$-groups


Author: Gary Thompson
Journal: Trans. Amer. Math. Soc. 341 (1994), 423-447
MSC: Primary 20C05; Secondary 16S34, 20C10, 20C11
DOI: https://doi.org/10.1090/S0002-9947-1994-1132878-2
MathSciNet review: 1132878
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Abstract: In 1986, Roggenkamp and Scott proved in [RS1]

Theorem 1.1. Let $ G$ be a finite $ p$-group for some prime $ p$, and $ S$ a local or semilocal Dedekind domain of characteristic 0 with a unique maximal ideal containing $ p$ (for example, $ S = {\mathbb{Z}_p}$ where $ {\mathbb{Z}_p}$ is the $ p$-adic integers). If $ H$ is a subgroup of the normalized units of $ SG$ with $ \vert H\vert = \vert G\vert$, then $ H$ is conjugate to $ G$ by an inner automorphism of $ SG$.

In the Appendix of a later paper [S], Scott outlined a possible proof of a related result:

Theorem 1.3. Let $ S$ be a complete, discrete valuation domain of characteristic 0 having maximal ideal $ \wp $ and residue field $ F \cong S/\wp $ of characteristic $ p$. Let $ G$ be a finite $ p$-group, and let $ U$ be a finite group of normalized units in $ SG$. Then there is a unit $ w$ in $ SG$ such that $ wU{w^{ - 1}} \leq G$.

The author later filled in that outline to give a complete proof of Theorem 1.3 and, at the urging of Scott, has been able to extend that result to

Theorem 1.2. Let $ S$ be a complete, discrete valuation ring of characteristic 0 having maximal ideal $ \wp $ containing $ p$. Let $ A$ be a local $ S$-algebra that is finitely generated as an $ S$-module, and let $ G$ be a finite $ p$-group. Then any finite, normalized subgroup of the $ S$-algebra $ \mathcal{A} = A{ \otimes _S}SG$ is conjugate to a subgroup of $ G$.


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DOI: https://doi.org/10.1090/S0002-9947-1994-1132878-2
Article copyright: © Copyright 1994 American Mathematical Society

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