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LOCALLY NILPOTENT p-GROUPS WHOSE PROPER SUBGROUPS ARE HYPERCENTRAL OR NILPOTENT-BY-CHERNIKOV

Published online by Cambridge University Press:  01 April 2000

A. O. ASAR
Affiliation:
Gazi Egitim Fakültesi, Gazi Üniversitesi, Teknikokullar 06500, Ankara, Turkey
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Abstract

Let G be a group and P be a property of groups. If every proper subgroup of G satisfies P but G itself does not satisfy it, then G is called a minimal non-P group. In this work we study locally nilpotent minimal non-P groups, where P stands for ‘hypercentral’ or ‘nilpotent-by-Chernikov’. In the first case we show that if G is a minimal non-hypercentral Fitting group in which every proper subgroup is solvable, then G is solvable (see Theorem 1.1 below). This result generalizes [3, Theorem 1]. In the second case we show that if every proper subgroup of G is nilpotent-by-Chernikov, then G is nilpotent-by-Chernikov (see Theorem 1.3 below). This settles a question which was considered in [1–3, 10]. Recently in [9], the non-periodic case of the above question has been settled but the same work contains an assertion without proof about the periodic case.

The main results of this paper are given below (see also [13]).

Type
Research Article
Copyright
The London Mathematical Society 2000

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