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Augmentation quotients of some nonabelian finite groups

Published online by Cambridge University Press:  24 October 2008

Gerald Losey  and
Nora Losey
Gerald Losey
Affiliation:
University of Manitoba, Winnipeg, Canada
Nora Losey
Affiliation:
University of Manitoba, Winnipeg, Canada
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Extract

1. Let G be a group, ZG its integral group ring and Δ = ΔG the augmentation ideal ZG By an augmentation quotient of G we mean any one of the ZG-modules

where n, r ≥ 1. In recent years there has been a great deal of interest in determining the abelian group structure of the augmentation quotients Qn(G) = Qn,1(G) and

(see (1, 2, 7, 8, 9, 12, 13, 14, 15)). Passi(8) has shown that in order to determine Qn(G) and Pn(G) for finite G it is sufficient to assume that G is a p-group. Passi(8, 9) and Singer(13, 14) have obtained information on the structure of these quotients for certain classes of abelian p-groups. However little seems to be known of a quantitative nature for nonabelian groups. In (2) Bachmann and Grünenfelder have proved the following qualitative result: if G is a finite group then there exist natural numbers n0 and π such that Qn(G) ≅ Qn+π (G) for all n ≥ n0; if Gω is the nilpotent residual of G and G/Gω has class c then π divides l.c.m. {1, 2, …, c}. There do not appear to be any examples in the literature of this periodic behaviour for c > 1. One of goals here is to present such examples. These examples will be from the class of finite p-groups in which the lower central series is an Np-series.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 85 , Issue 2 , March 1979 , pp. 261 - 270
Copyright
Copyright © Cambridge Philosophical Society 1979

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References

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