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Generalizing a theorem of P. Hall on finite-by-nilpotent groups

Authors: Gustavo A. Fernández-Alcober and Marta Morigi
Journal: Proc. Amer. Math. Soc. 137 (2009), 425-429
MSC (2000): Primary 20F14
Published electronically: September 15, 2008
MathSciNet review: 2448560
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Abstract: Let $\gamma _i(G)$ and $Z_i(G)$ denote the $i$-th terms of the lower and upper central series of a group $G$, respectively. In 1956 P. Hall showed that if $\gamma _{i+1}(G)$ is finite, then the index $|G:Z_{2i}(G)|$ is finite. We prove that the same result holds under the weaker hypothesis that $|\gamma _{i+1}(G):\gamma _{i+1}(G)\cap Z_i(G)|$ is finite.

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Additional Information

Gustavo A. Fernández-Alcober
Affiliation: Matematika Saila, Euskal Herriko Unibertsitatea, 48080 Bilbao, Spain
MR Author ID: 307028

Marta Morigi
Affiliation: Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, 40127 Bologna, Italy

Keywords: Upper and lower central series; Finite-by-nilpotent groups
Received by editor(s): November 7, 2007
Published electronically: September 15, 2008
Additional Notes: The first author is supported by the Spanish Ministry of Science and Education, grant MTM2004-04665, partly with FEDER funds, and by the Basque Government, grant IT-252-07.
The second author is partially supported by MIUR (Project “Teoria dei Gruppi e applicazioni”) and thanks the University of the Basque Country for its hospitality.
Communicated by: Jonathan I. Hall
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.