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

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

*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**
—, - 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, - 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
—, - 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
—, - 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, - 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,

*Methods of representation theory*, vol. 1, Wiley Interscience, 1962.

*Units in group rings*, D. Phil, thesis, Oxford Univ., 1940.

*Subgroup rigidity of*$p$-

*adic group rings*, preprint, June, 1989.

*On a conjecture of Zassenhaus for finite group rings*, preprint (submitted).

*Subgroup rigidity in group rings*, Ph.D. thesis, Univ. of Virginia, 1990.

*The group ring problem*, Ph.D. thesis, Univ. of Chicago, 1968.

## 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