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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the upper central series of infinite groups


Authors: M. De Falco, F. de Giovanni, C. Musella and Y. P. Sysak
Journal: Proc. Amer. Math. Soc. 139 (2011), 385-389
MSC (2010): Primary 20E15
Published electronically: September 27, 2010
MathSciNet review: 2736322
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Abstract: Two relevant theorems by R. Baer and P. Hall show that a group is finite over a term with finite ordinal type of its upper central series if and only if it is finite-by-nilpotent. Extending these results, we prove here that if $ G$ is any group, the hypercentre factor group $ G/\bar Z(G)$ is finite if and only if $ G$ contains a finite normal subgroup $ N$ such that $ G/N$ is hypercentral (where the hypercentre $ \bar Z(G)$ of $ G$ is defined as the last term of its upper central series).


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Additional Information

M. De Falco
Affiliation: Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, Complesso Universitario Monte S. Angelo, Via Cintia, I-80126 Napoli, Italy
Email: mdefalco@unina.it

F. de Giovanni
Affiliation: Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, Complesso Universitario Monte S. Angelo, Via Cintia, I-80126 Napoli, Italy
Email: degiovan@unina.it

C. Musella
Affiliation: Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, Complesso Universitario Monte S. Angelo, Via Cintia, I-80126 Napoli, Italy
Email: cmusella@unina.it

Y. P. Sysak
Affiliation: Institute of Mathematics, Ukrainian National Academy of Sciences, vul. Tereshchenkivska 3, 01601 Kiev, Ukraine
Email: sysak@imath.kiev.ua

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10625-1
PII: S 0002-9939(2010)10625-1
Keywords: Upper central series, hypercentre, hypercentral group
Received by editor(s): November 6, 2009
Published electronically: September 27, 2010
Dedicated: To Bernhard Amberg on his 70th birthday
Communicated by: Jonathan I. Hall
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.