Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Subgroup rigidity in finite-dimensional group algebras over $p$-groups
HTML articles powered by AMS MathViewer

by Gary Thompson
Trans. Amer. Math. Soc. 341 (1994), 423-447
DOI: https://doi.org/10.1090/S0002-9947-1994-1132878-2

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 $|H| = |G|$, 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 $G$.
References
  • 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
  • —, Methods of representation theory, vol. 1, Wiley Interscience, 1962.
  • Everett C. Dade, Deux groupes finis distincts ayant la même algèbre de groupe sur tout corps, Math. Z. 119 (1971), 345–348 (French). MR 280610, DOI 10.1007/BF01109886
  • G. Higman, Units in group rings, D. Phil, thesis, Oxford Univ., 1940.
  • Thomas W. Hungerford, Algebra, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1974. MR 0354211
  • Gregory Karpilovsky, Unit groups of classical rings, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1988. MR 978631
  • 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
  • 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, DOI 10.1090/pspum/047.2/933437
  • —, Subgroup rigidity of $p$-adic group rings, preprint, June, 1989.
  • 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
  • Klaus Roggenkamp and Leonard Scott, Isomorphisms of $p$-adic group rings, Ann. of Math. (2) 126 (1987), no. 3, 593–647. MR 916720, DOI 10.2307/1971362
  • —, On a conjecture of Zassenhaus for finite group rings, preprint (submitted).
  • 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, DOI 10.1090/pspum/047.1/933364
  • Sudarshan K. Sehgal, Topics in group rings, Monographs and Textbooks in Pure and Applied Mathematics, vol. 50, Marcel Dekker, Inc., New York, 1978. MR 508515
  • G. Thompson, Subgroup rigidity in group rings, Ph.D. thesis, Univ. of Virginia, 1990.
  • Alfred Weiss, Rigidity of $p$-adic $p$-torsion, Ann. of Math. (2) 127 (1988), no. 2, 317–332. MR 932300, DOI 10.2307/2007056
  • Edwin Weiss, Algebraic number theory, McGraw-Hill Book Co., Inc., New York-San Francisco-Toronto-London, 1963. MR 0159805
  • A. Whitcomb, The group ring problem, Ph.D. thesis, Univ. of Chicago, 1968.
Similar Articles
Bibliographic Information
  • © Copyright 1994 American Mathematical Society
  • 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