Subgroup rigidity in finitedimensional group algebras over groups
Author:
Gary Thompson
Journal:
Trans. Amer. Math. Soc. 341 (1994), 423447
MSC:
Primary 20C05; Secondary 16S34, 20C10, 20C11
MathSciNet review:
1132878
Fulltext PDF Free Access
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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 .
 [CR1]
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MR
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 [CR1]
 C. W. Curtis and I. Reiner, Representation theory of finite groups and associative algebras, Wiley Interscience, 1962. MR 1013113 (90g:16001)
 [CR2]
 , Methods of representation theory, vol. 1, Wiley Interscience, 1962.
 [D]
 E. C. Dade, Deux groups finis ayant le même algèbre de groupe sur tout corps, Math. Z. 119 (1971). MR 0280610 (43:6329)
 [Hi]
 G. Higman, Units in group rings, D. Phil, thesis, Oxford Univ., 1940.
 [Hu]
 T. W. Hungerford, Algebra, Holt, Rhinehart, and Winston, 1974. MR 0354211 (50:6693)
 [K]
 G. Karpilovsky, Unit groups of classical rings, Oxford Univ. Press, 1988. MR 978631 (90e:20007)
 [M]
 H. Matsumura, Commutative ring theory, Cambridge Univ. Press, 1986. MR 879273 (88h:13001)
 [R1]
 K. W. Roggenkamp, Picard groups of integral group rings of nilpotent groups, Proc. Sympos. Pure Math., vol. 47, Amer. Math. Soc., Providence, R.I., 1987, pp. 477486. MR 933437 (89e:20013)
 [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, Proc. of GroupsSt. Andrews 1985, Cambridge Univ. Press, 1986. MR 896526 (88e:20010)
 [RS2]
 , Isomorphisms of adic group rings, Ann. of Math. (2) 126 (1987), 593647. MR 916720 (89b:20021)
 [RS3]
 , On a conjecture of Zassenhaus for finite group rings, preprint (submitted).
 [S]
 L. L. Scott, Recent progress on the isomorphism problem, Proc. Sympos. Pure Math., vol. 47, Amer. Math. Soc., Providence, R.I., 1987, pp. 259274. MR 933364 (89c:20015)
 [Se]
 S. K.. Sehgal, Topics in group rings, Marcel Dekker, 1978. MR 508515 (80j:16001)
 [T]
 G. Thompson, Subgroup rigidity in group rings, Ph.D. thesis, Univ. of Virginia, 1990.
 [W]
 A. Weiss, Rigidity of adic torsion, Ann. of Math. (2) 127 (1988), 317332. MR 932300 (89g:20010)
 [We]
 E. Weiss, Algebraic number theory, McGrawHill. MR 0159805 (28:3021)
 [Wh]
 A. Whitcomb, The group ring problem, Ph.D. thesis, Univ. of Chicago, 1968.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199411328782
PII:
S 00029947(1994)11328782
Article copyright:
© Copyright 1994
American Mathematical Society
