Unit groups of completed modular group algebras and the isomorphism problem

Author:
Frank Röhl

Journal:
Proc. Amer. Math. Soc. **111** (1991), 611-618

MSC:
Primary 16U60; Secondary 16W10, 16W50, 20C05

DOI:
https://doi.org/10.1090/S0002-9939-1991-1045598-2

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 Lie--algebra 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 .

**[1]**L. R. Ivory,*A note on normal complements in**envelopes*, Proc. Amer. Math. Soc.**79**(1980), 9-12. MR**560574 (82e:20004)****[2]**D. L. Johnson,*The modular group ring of a finite**-group*, Proc. Amer. Math. Soc.**68**(1978), 19-22. MR**0457539 (56:15744)****[3]**V. M. Levchuk,*Relation of the unitriangular group to certain rings*, Algebra and Logic**15**(1976), 348-360. (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), 331-338. MR**0447403 (56:5715)****[6]**I. B. S. Passi,*Group rings and their augmentation ideals*, Lecture Notes in Math., vol. 715, Springer-Verlag, New York, 1979. MR**537126 (80k:20009)****[7]**D. Quillen,*On the associated graded ring of a group ring*, J. Algebra**10**(1968), 411-418. MR**0231919 (38:245)****[8]**K. W. Roggenkamp and L. Scott,*The isomorphism problem and units in group rings of finite order*, in Groups-St. Andrews 1981, London Math. Soc. Lecture Notes, vol. 71, 1982, pp. 313-327.**[9]**F. Rõhl,*On induced isomorphisms of group rings*, in Groups-Korea 1983, Lecture Notes in Math., vol. 1098, Springer-Verlag, New York, 1984, pp. 136-141. MR**781366 (88h:20008)****[10]**R. Sandling,*Group rings of circle and unit groups*, Math. Z.**140**(1974), 195-202. MR**0382332 (52:3217)****[11]**-,*The isomorphism problem for group rings: a survey*, in Orders and Their Applications, Lecture Notes in Math., vol. 1142, Springer-Verlag, New York, 1985, pp. 256-288. MR**812504 (87b:20007)****[12]**-,*The modular group algebra of a central-elementary-by-abelian*-group, Arch. Math.**52**(1989), 22-27. 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, 1960-61.

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DOI:
https://doi.org/10.1090/S0002-9939-1991-1045598-2

Article copyright:
© Copyright 1991
American Mathematical Society