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HALL CLASSES OF GROUPS WITH A LOCALLY FINITE OBSTRUCTION

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  24 July 2023

F. DE GIOVANNI Open the ORCID record for F. DE GIOVANNI [Opens in a new window] ,
M. TROMBETTI Open the ORCID record for M. TROMBETTI [Opens in a new window]  and
B. A. F. WEHRFRITZ
F. DE GIOVANNI*
Affiliation:
Dipartimento di Matematica e Appl., Università di Napoli Federico II, Napoli, Italy
M. TROMBETTI
Affiliation:
Dipartimento di Matematica e Appl., Università di Napoli Federico II, Napoli, Italy e-mail: marco.trombetti@unina.it
B. A. F. WEHRFRITZ
Affiliation:
School of Mathematical Sciences, Queen Mary University of London, London, UK e-mail: b.a.f.wehrfritz@qmul.ac.uk
*
e-mail: degiovan@unina.it
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Abstract

A well-known theorem of Philip Hall states that if a group G has a nilpotent normal subgroup N such that $G/N'$ is nilpotent, then G itself is nilpotent. We say that a group class 𝔛 is a Hall class if it contains every group G admitting a nilpotent normal subgroup N such that $G/N'$ belongs to 𝔛. Hall classes have been considered by several authors, such as Plotkin [‘Some properties of automorphisms of nilpotent groups’, Soviet Math. Dokl. 2 (1961), 471–474] and Robinson [‘A property of the lower central series of a group’, Math. Z. 107 (1968), 225–231]. A further detailed study of Hall classes is performed by us in another paper [‘Hall classes of groups’, to appear] and we also investigate the behaviour of the class of finite-by-𝔜 groups for a given Hall class 𝔜 [‘Hall classes in linear groups’, to appear]. The aim of this paper is to prove that for most natural choices of the Hall class 𝔜, also the classes $(\mathbf{L}\mathfrak{F})\mathfrak{Y}$ and 𝔅𝔜 are Hall classes, where L𝔉 is the class of locally finite groups and 𝔅 is the class of locally finite groups of finite exponent.

Keywords

Hall classnilpotent grouplocally finite group

MSC classification

Primary: 20F18: Nilpotent groups
Secondary: 20F19: Generalizations of solvable and nilpotent groups 20F50: Periodic groups; locally finite groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , First View , pp. 1 - 28
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by Benjamin Martin

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