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Groupings of Metabelian Groups and Extension Categories

Published online by Cambridge University Press:  20 November 2018

K. W. Roggenkamp
K. W. Roggenkamp*
Affiliation:
Universität Stuttgart, Stuttgart, West Germany
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Let G be a metabelian group and R an integral domain of characteristic zero, such that no rational prime divisor of │G│ is invertible in R. By RG we denote the group ring of G over R. In this note we shall prove

THEOREM. If RG ≌ RH as R-algebras, then G ≌ H

The question whether this result holds was posed to me by S. K. Sehgal. The result for R = Z is contained in G. Higman's thesis, and he apparently also proved a more general result. At any rate, I think that the methods of the proof are interesting eo ipso, since they establish a “Noether-Deuring theorem” for extension categories.

In proving the above result, it is necessary to study closely the category of extensions (ℊs, S), where the objects are short exact sequences of SG-modules

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 2 , February 1980 , pp. 449 - 459
Copyright
Copyright © Canadian Mathematical Society 1980

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