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 | References | Similar Articles | Additional Information

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* .

**[CR1]**Charles W. Curtis and Irving Reiner,*Representation theory of finite groups and associative algebras*, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. Reprint of the 1962 original; A Wiley-Interscience Publication. MR**1013113****[CR2]**-,*Methods of representation theory*, vol. 1, Wiley Interscience, 1962.**[D]**Everett C. Dade,*Deux groupes finis distincts ayant la même algèbre de groupe sur tout corps*, Math. Z.**119**(1971), 345–348 (French). MR**0280610**, https://doi.org/10.1007/BF01109886**[Hi]**G. Higman,*Units in group rings*, D. Phil, thesis, Oxford Univ., 1940.**[Hu]**Thomas W. Hungerford,*Algebra*, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1974. MR**0354211****[K]**Gregory Karpilovsky,*Unit groups of classical rings*, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1988. MR**978631****[M]**Hideyuki Matsumura,*Commutative ring theory*, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR**879273****[R1]**K. W. Roggenkamp,*Picard groups of integral group rings of nilpotent groups*, The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986) Proc. Sympos. Pure Math., vol. 47, Amer. Math. Soc., Providence, RI, 1987, pp. 477–485. MR**933437****[R2]**-,*Subgroup rigidity of*-*adic group rings*, preprint, June, 1989.**[RS1]**K. W. Roggenkamp and L. L. Scott,*The isomorphism problem for integral group rings of finite nilpotent groups*, Proceedings of groups—St. Andrews 1985, London Math. Soc. Lecture Note Ser., vol. 121, Cambridge Univ. Press, Cambridge, 1986, pp. 291–299. MR**896526****[RS2]**Klaus Roggenkamp and Leonard Scott,*Isomorphisms of 𝑝-adic group rings*, Ann. of Math. (2)**126**(1987), no. 3, 593–647. MR**916720**, https://doi.org/10.2307/1971362**[RS3]**-,*On a conjecture of Zassenhaus for finite group rings*, preprint (submitted).**[S]**Leonard L. Scott,*Recent progress on the isomorphism problem*, The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986) Proc. Sympos. Pure Math., vol. 47, Amer. Math. Soc., Providence, RI, 1987, pp. 259–273. MR**933364****[Se]**Sudarshan K. Sehgal,*Topics in group rings*, Monographs and Textbooks in Pure and Applied Math., vol. 50, Marcel Dekker, Inc., New York, 1978. MR**508515****[T]**G. Thompson,*Subgroup rigidity in group rings*, Ph.D. thesis, Univ. of Virginia, 1990.**[W]**Alfred Weiss,*Rigidity of 𝑝-adic 𝑝-torsion*, Ann. of Math. (2)**127**(1988), no. 2, 317–332. MR**932300**, https://doi.org/10.2307/2007056**[We]**Edwin Weiss,*Algebraic number theory*, McGraw-Hill Book Co., Inc., New York-San Francisco-Toronto-London, 1963. MR**0159805****[Wh]**A. Whitcomb,*The group ring problem*, Ph.D. thesis, Univ. of Chicago, 1968.

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

Article copyright:
© Copyright 1994
American Mathematical Society