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Dividierbarkeit und Typenbedingungen in verallgemeinerten nilpotenten Gruppen

Divisibilty and type-conditions in generalized nilpotent groups

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Abstract

In this paper we generalize theorems of S. N. Černikov, G. Baumslag and R. B. Warfield. S. N. Černikov showed that a hypercentral groupG is π-divisible whenever,G/G' is π-divisible. His theorem is identical to a special case of TheoremA. G. Baumslag andR. B. Warfield proved that the commutator subgroupΓ k of a nilpotent groupG is π-divisible wheneverG (Baumslag [1], 14.5) orG/Z k-1 (Warfield [8], 4.13) is π-divisible. We show this implication for hypercentral and the torsion-free weakly nilpotent groups. In TheoremD similar statements for the typesets ofG/Z k-1 andΓ k are proved; hereG is nilpotent or torsion-free weakly nilpotent. The commutator formulas of Sect. 3 are essential for the proof of the theorems. These formulas seem to be new and are interesting in their own right.

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Authors and Affiliations

  1. Mathematisches Institut der Universität Würzburg, Am Hubland, D-8700, Würzburg, Deutschland

    Hans Peter Heislbetz

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  1. Hans Peter Heislbetz
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Heislbetz, H.P. Dividierbarkeit und Typenbedingungen in verallgemeinerten nilpotenten Gruppen. Monatshefte für Mathematik 115, 67–81 (1993). https://doi.org/10.1007/BF01311211

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  • DOI: https://doi.org/10.1007/BF01311211

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