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Augmentation quotients of group rings and symmetric powers

Published online by Cambridge University Press:  24 October 2008

Robert Sandling  and
Ken-Ichi Tahara
Robert Sandling
Affiliation:
The University, Manchester, M13 9PL, England
Ken-Ichi Tahara
Affiliation:
Aichi University of Education, Igaya-cho, Kariya-shi, 448, Japan
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Extract

Let G be a group with the lower central series

Let

where Σ runs over all non-negative integers a1, a2,…, an such that and is the aith symmetric power of the abelian group Gi/Gi+1 where

Let I (G) be the augmentation ideal of G in , the group ring of G over . Define the additive group Qn (G) = In (G) / In+1 (G) for any n ≥ 1. Then it is well known that Q1(G) ≅ W1(G) for any group G. Losey (4,5) proved that Q2(G) ≅ W2(G) for any finitely generated group G. Furthermore recently Tahara(12) proved that Q3(G) is a certain precisely defined quotient of W3(G) for any finite group G.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 85 , Issue 2 , March 1979 , pp. 247 - 252
Copyright
Copyright © Cambridge Philosophical Society 1979

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References

REFERENCES

(1)Grün, O.Über eine Faktorgruppe freier Gruppen, I. Deutsche Math. 1 (1936), 772–782.Google Scholar
(2)Hall, P.Nilpotent groups. Canad. Math. Congress, Univ. of Alberta, 1957; Queen Mary College Math. Notes, 1969.Google Scholar
(3)Hartley, B.The residual nilpotence of wreath products. Proc. London Math. Soc. (3) 20 (1970), 365–392.Google Scholar
(4)Losey, G.On the structure of Q 2(G) for finitely generated groups. Canad. J. Math. 25 (1973), 353–359.CrossRefGoogle Scholar
(5)Losey, G.N-series and filtrations of the augmentation ideal. Canad. J. Math. 26 (1974), 962–977.Google Scholar
(6)MaCmahon, P. A.Combinatory analysis, vol. I (Cambridge, Cambridge University Press, 1915).Google Scholar
(7)Magnus, W.¨Uber Beziehungen zwischen höheren Kommutatoren. J. Reine Angew. Math. 177 (1937), 105–115.CrossRefGoogle Scholar
(8)Passi, I. B. S.Polynomial maps on groups. J. Algebra 9 (1968), 121–151.CrossRefGoogle Scholar
(9)Passi, I. B. S.Polynomial functors. Proc. Cambridge Philos. Soc. 66 (1969), 505–512.Google Scholar
(10)Quillen, D. G.On the associated graded ring of a group ring. J. Algebra 10 (1968), 411–418.CrossRefGoogle Scholar
(11)Sandling, R.Dimension subgroups over arbitrary coefficient rings. J. Algebra 21 (1972), 250–265.CrossRefGoogle Scholar
(12)Tahara, K.On the structure of Q 3(G) and the fourth dimension subgroups. Japan. J. Math. (N.S.) 3 (1977), 381–394.Google Scholar
(13)Tahara, K. The augmentation quotients of group rings and the fifth dimension subgroups. (To appear).Google Scholar