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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Generalizing a theorem of P. Hall on finite-by-nilpotent groups
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by Gustavo A. Fernández-Alcober and Marta Morigi PDF
Proc. Amer. Math. Soc. 137 (2009), 425-429 Request permission


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
  • Email:
  • Marta Morigi
  • Affiliation: Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, 40127 Bologna, Italy
  • Email:
  • 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
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 137 (2009), 425-429
  • MSC (2000): Primary 20F14
  • DOI:
  • MathSciNet review: 2448560