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

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]**Lee R. Ivory,*A note on normal complements in mod 𝑝 envelopes*, Proc. Amer. Math. Soc.**79**(1980), no. 1, 9–12. MR**560574**, 10.1090/S0002-9939-1980-0560574-4**[2]**D. L. Johnson,*The modular group-ring of a finite 𝑝-group*, Proc. Amer. Math. Soc.**68**(1978), no. 1, 19–22. MR**0457539**, 10.1090/S0002-9939-1978-0457539-0**[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 𝑚𝑜𝑑𝑝-envelopes*, Israel J. Math.**27**(1977), no. 3-4, 331–338. MR**0447403****[6]**Inder Bir S. Passi,*Group rings and their augmentation ideals*, Lecture Notes in Mathematics, vol. 715, Springer, Berlin, 1979. MR**537126****[7]**Daniel G. Quillen,*On the associated graded ring of a group ring*, J. Algebra**10**(1968), 411–418. MR**0231919****[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]**Frank Röhl,*On induced isomorphisms of group rings*, Groups—Korea 1983 (Kyoungju, 1983) Lecture Notes in Math., vol. 1098, Springer, Berlin, 1984, pp. 136–141. MR**781366**, 10.1007/BFb0099670**[10]**Robert Sandling,*Group rings of circle and unit groups*, Math. Z.**140**(1974), 195–202. MR**0382332****[11]**Robert Sandling,*The isomorphism problem for group rings: a survey*, Orders and their applications (Oberwolfach, 1984) Lecture Notes in Math., vol. 1142, Springer, Berlin, 1985, pp. 256–288. MR**812504**, 10.1007/BFb0074806**[12]**Robert Sandling,*The modular group algebra of a central-elementary-by-abelian 𝑝-group*, Arch. Math. (Basel)**52**(1989), no. 1, 22–27. MR**980047**, 10.1007/BF01197966**[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:
http://dx.doi.org/10.1090/S0002-9939-1991-1045598-2

Article copyright:
© Copyright 1991
American Mathematical Society