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NILPOTENCY IN UNCOUNTABLE GROUPS

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  27 October 2016

FRANCESCO DE GIOVANNI  and
MARCO TROMBETTI
FRANCESCO DE GIOVANNI*
Affiliation:
Dip. di Matematica e Appl., Università di Napoli Federico II, Via Cintia, Napoli, Italy email [email protected]
MARCO TROMBETTI
Affiliation:
Dip. di Matematica e Appl., Università di Napoli Federico II, Via Cintia, Napoli, Italy email [email protected]
*
✉[email protected]
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Abstract

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The main purpose of this paper is to investigate the behaviour of uncountable groups of cardinality $\aleph$ in which all proper subgroups of cardinality $\aleph$ are nilpotent. It is proved that such a group $G$ is nilpotent, provided that $G$ has no infinite simple homomorphic images and either $\aleph$ has cofinality strictly larger than $\aleph _{0}$ or the generalized continuum hypothesis is assumed to hold. Furthermore, groups whose proper subgroups of large cardinality are soluble are studied in the last part of the paper.

Keywords

uncountable groupnilpotent groupsoluble group

MSC classification

Primary: 20F18: Nilpotent groups
Secondary: 20F19: Generalizations of solvable and nilpotent groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 103 , Issue 1 , August 2017 , pp. 59 - 69
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

Footnotes

The authors are members of GNSAGA-INdAM, and this work was carried out within the ADV-AGTA project.

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