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Parametrizing Ω1(Z): Singular Case

  • F. E. A. JohnsonEmail author
Chapter
  • 921 Downloads
Part of the Algebra and Applications book series (AA, volume 17)

Abstract

In this chapter we work under the blanket assumption that G is a finitely generated group with abelianization G ab and integral group ring Λ=Z[G], and that \(\operatorname{Ext}_{\varLambda}^{1}(\mathbf{ Z}, \varLambda)\neq0\). Then G is necessarily infinite. We investigate minimality of \(\mathcal{I}\) in Ω 1(Z) and first establish:
Second minimality criterion:

\(\mathcal{I}\) lies at the minimal level of Ω 1(Z) if G ab is finite.

This second criterion also applies to many cases where \(\operatorname {Ext}_{\varLambda}^{1}(\mathbf{ Z}, \varLambda) = 0\) although we do not need to use it there. We employ it in Sect. 14.3 to give examples of groups G with infinite splitting in Ω 1(Z).

Keywords

Split Infinitives Integral Group Ring Blanket Assumption Minimality Criterion Nontrivial Finite Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 41.
    Higgins, P.J.: Categories and Groupoids. Mathematical Studies. Van Nostrand/Reinhold, Princeton/New York (1971) zbMATHGoogle Scholar
  2. 60.
    Johnson, F.E.A., Wall, C.T.C.: On groups satisfying Poincaré Duality. Ann. Math. 96, 592–598 (1972) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity College LondonLondonUK

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