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A Universal Coefficient Decomposition for Subgroups Induced by Submodules of Group Algebras

Published online by Cambridge University Press:  20 November 2018

Manfred Hartl
Manfred Hartl*
Affiliation:
U.R.A. au C.N.R.S. 751, et Département de Mathématiques Université de Valenciennes le Mont Houy, BP 311 59304 Valenciennes Cedex France
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Abstract

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Dimension subgroups and Lie dimension subgroups are known to satisfy a ‘universal coefficient decomposition’, i.e. their value with respect to an arbitrary coefficient ring can be described in terms of their values with respect to the ‘universal’ coefficient rings given by the cyclic groups of infinite and prime power order. Here this fact is generalized to much more general types of induced subgroups, notably covering Fox subgroups and relative dimension subgroups with respect to group algebra filtrations induced by arbitrary N-series, as well as certain common generalisations of these which occur in the study of the former. This result relies on an extension of the principal universal coefficient decomposition theorem on polynomial ideals (due to Passi, Parmenter and Seghal), to all additive subgroups of group rings. This is possible by using homological instead of ring theoretical methods.

Keywords

Primary: 20C07secondary: 16A27induced subgroupsgroup algebrasFox subgroupsrelative dimension subgroupspolynomial ideals
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 40 , Issue 1 , 01 March 1997 , pp. 47 - 53
Copyright
Copyright © Canadian Mathematical Society 1997

References

1. Hartl, M., Une approche homologique des sous-groupes et quotients de Fox, C. R. Acad. Sci. Paris, to appear.Google Scholar
2. Hartl, M., The third relative Fox dimension subgroup, submitted.Google Scholar
3. Hartl, M., The second Fox quotient of arbitrary group rings, preprint.Google Scholar
4. Lane, S. Mac, Homology, Springer Grundlehren 114, Springer-Verlag, Berlin, Göttingen, Heidelberg, 1963.Google Scholar
5. Parmenter, M. M., I. B. S., Passi and Sehgal, S. K., Polynomial ideals in group rings, Canad. J. Math. 25 (1973), 1174–1182.Google Scholar
6. I. B. S., Passi, Group Rings and Their Augmentation Ideals, Lecture Notes inMath. 715, Springer-Verlag, Berlin, Heidelberg, New York, 1979.Google Scholar
7. Sandling, R., Dimension subgroups over arbitrary coefficient rings, J. Algebra 21 (1972), 250–265.Google Scholar
8. Sandling, R., Note on the integral group ring problem, Math. Z. 124 (1972), 255–258.Google Scholar