Subgroup rigidity in finite-dimensional group algebras over -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* *be a finite* -*group for some prime* , *and* *a local or semilocal Dedekind domain of characteristic* 0 *with a unique maximal ideal containing* (*for example*, *where* *is the* -*adic integers*). *If* *is a subgroup of the normalized units of* *with* , *then* *is conjugate to* *by an inner automorphism of* .

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

**Theorem 1.3.** *Let* *be a complete, discrete valuation domain of characteristic* 0 *having maximal ideal* *and residue field* *of characteristic* . *Let* *be a finite* -*group, and let* *be a finite group of normalized units in* . *Then there is a unit* *in* *such that* .

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* *be a complete, discrete valuation ring of characteristic* 0 *having maximal ideal* *containing* . *Let* *be a local* -*algebra that is finitely generated as an* -*module, and let* *be a finite* -*group. Then any finite, normalized subgroup of the* -*algebra* *is conjugate to a subgroup of* .

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DOI:
https://doi.org/10.1090/S0002-9947-1994-1132878-2

Article copyright:
© Copyright 1994
American Mathematical Society