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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

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