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On a generalization of Baer Theorem


Authors: L. A. Kurdachenko, J. Otal and I. Ya. Subbotin
Journal: Proc. Amer. Math. Soc. 141 (2013), 2597-2602
MSC (2010): Primary 20F14
Published electronically: April 9, 2013
MathSciNet review: 3056549
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Abstract: R. Baer has proved that if the factor-group $ G/\zeta _n(G)$ of a group $ G$ by the member $ \zeta _n(G)$ of its upper central series is finite (here $ n$ is a positive integer), then the member $ \gamma _{n+1}(G)$ of the lower central series of $ G$ is also finite. In particular, in this case, the nilpotent residual of $ G$ is finite. This theorem admits the following simple generalization, which has been published very recently by M. de Falco, F. de Giovanni, C. Musella and Ya. P. Sysak: ``If the factor-group $ G/Z$ of a group $ G$ modulo its upper hypercenter $ Z$ is finite, then G has a finite normal subgroup $ L$ such that $ G/L$ is hypercentral.'' In the current article we offer a new, simpler, very short proof of this theorem and specify it substantially. In fact, we prove that if $ \vert G/Z\vert = t$, then $ \vert L\vert\leq t^k$, where $ k = \frac {1}{2}(log_pt+1)$ and $ p$ is the least prime divisor of $ t$.


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Additional Information

L. A. Kurdachenko
Affiliation: Department of Algebra, National University of Dnepropetrovsk, Vul. Naukova 13, Dnepropetrovsk 50, Ukraine 49050
Address at time of publication: Department of Algebra, School of Mathematics and Mechanics, National University of Dnepropetrovsk, Gagarin Prospect 72, Dnepropetrovsk 10, 49010 Ukraine
Email: lkurdachenko@gmail.com

J. Otal
Affiliation: Department of Mathematics - IUMA, University of Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain
Email: otal@unizar.es

I. Ya. Subbotin
Affiliation: Department of Mathematics and Natural Sciences, National University, 5245 Pacific Concourse Drive, Los Angeles, California 90045
Email: isubboti@nu.edu

DOI: https://doi.org/10.1090/S0002-9939-2013-11677-1
Received by editor(s): October 31, 2011
Published electronically: April 9, 2013
Additional Notes: The authors were supported by Proyecto MTM2010-19938-C03-03 of MICINN (Spain), the Government of Aragón (Spain) and FEDER funds from the European Union
Communicated by: Ken Ono
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.