Parametrizing Ω1(Z): Singular Case

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


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).


Split Infinitives Integral Group Ring Blanket Assumption Minimality Criterion Nontrivial Finite Group 
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Copyright information

© Springer-Verlag London Limited 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity College LondonLondonUK

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