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Archiv der Mathematik

, Volume 96, Issue 1, pp 31–37 | Cite as

Lattice-theoretic characterizations of the group classes \({\mathfrak{N}^{k-1}\mathfrak{A}}\) and \({\mathfrak{N}^k}\) for k ≥ 2

  • Roland Schmidt
Article

Abstract

Let \({2\leq k\in \mathbb{N}}\). Recently, Costantini and Zacher obtained a lattice-theoretic characterization of the classes \({\mathfrak{N}^k}\) of finite soluble groups with nilpotent length at most k. It is the aim of this paper to give a lattice-theoretic characterization of the classes \({\mathfrak{N}^{k-1}\mathfrak{A}}\) of finite groups with commutator subgroup in \({\mathfrak{N}^{k-1}}\); in addition, our method also yields a new characterization of the classes \({\mathfrak{N}^k}\). The main idea of our approach is to use two well-known theorems of Gaschütz on the Frattini and Fitting subgroups of finite groups.

Mathematics Subject Classification (2000)

20D30 

Keywords

Lattice-theoretic characterization Nilpotent-by-abelian groups 

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Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.Mathematisches SeminarUniversität KielKielGermany

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