Unit groups of completed modular group algebras and the isomorphism problem
Author:
Frank Röhl
Journal:
Proc. Amer. Math. Soc. 111 (1991), 611618
MSC:
Primary 16U60; Secondary 16W10, 16W50, 20C05
MathSciNet review:
1045598
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Abstract: In this paper it is shown that isomorphism of group algebras of finite groups over the field of elements implies isomorphism of the groups, if one of the groups has a normal complement in the group of normalized units of the group algebra. Furthermore, a class of groups satisfying this condition is provided, and it is shown that the associated graded Liealgebra of the group of normalized units of the Magnus algebra of being a residually 'nilpotent group of bounded exponent', is a split extension of the one associated to .
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 [1]
 L. R. Ivory, A note on normal complements in envelopes, Proc. Amer. Math. Soc. 79 (1980), 912. MR 560574 (82e:20004)
 [2]
 D. L. Johnson, The modular group ring of a finite group, Proc. Amer. Math. Soc. 68 (1978), 1922. MR 0457539 (56:15744)
 [3]
 V. M. Levchuk, Relation of the unitriangular group to certain rings, Algebra and Logic 15 (1976), 348360. (translation)
 [4]
 L. E. Moran, The modular group ring of a group, M. Phil. Thesis, University of Nottingham, 1972.
 [5]
 L. E. Moran and R. N. Tench, Normal complements in envelopes, Israel J. Math. 27 (1977), 331338. MR 0447403 (56:5715)
 [6]
 I. B. S. Passi, Group rings and their augmentation ideals, Lecture Notes in Math., vol. 715, SpringerVerlag, New York, 1979. MR 537126 (80k:20009)
 [7]
 D. Quillen, On the associated graded ring of a group ring, J. Algebra 10 (1968), 411418. MR 0231919 (38:245)
 [8]
 K. W. Roggenkamp and L. Scott, The isomorphism problem and units in group rings of finite order, in GroupsSt. Andrews 1981, London Math. Soc. Lecture Notes, vol. 71, 1982, pp. 313327.
 [9]
 F. Rõhl, On induced isomorphisms of group rings, in GroupsKorea 1983, Lecture Notes in Math., vol. 1098, SpringerVerlag, New York, 1984, pp. 136141. MR 781366 (88h:20008)
 [10]
 R. Sandling, Group rings of circle and unit groups, Math. Z. 140 (1974), 195202. MR 0382332 (52:3217)
 [11]
 , The isomorphism problem for group rings: a survey, in Orders and Their Applications, Lecture Notes in Math., vol. 1142, SpringerVerlag, New York, 1985, pp. 256288. MR 812504 (87b:20007)
 [12]
 , The modular group algebra of a centralelementarybyabelian group, Arch. Math. 52 (1989), 2227. MR 980047 (90b:20007)
 [13]
 H. N. Ward, Some results on the group algebra of a group over a prime field, Seminar on Finite Groups and Related Topics at Harvard University, 196061.
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DOI:
http://dx.doi.org/10.1090/S00029939199110455982
PII:
S 00029939(1991)10455982
Article copyright:
© Copyright 1991 American Mathematical Society
