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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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On locally finite $p$-groups satisfying an Engel condition
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by Alireza Abdollahi and Gunnar Traustason PDF
Proc. Amer. Math. Soc. 130 (2002), 2827-2836 Request permission


For a given positive integer $n$ and a given prime number $p$, let $r=r(n,p)$ be the integer satisfying $p^{r-1}<n\leq p^{r}$. We show that every locally finite $p$-group, satisfying the $n$-Engel identity, is (nilpotent of $n$-bounded class)-by-(finite exponent) where the best upper bound for the exponent is either $p^{r}$ or $p^{r-1}$ if $p$ is odd. When $p=2$ the best upper bound is $p^{r-1},p^{r}$ or $p^{r+1}$. In the second part of the paper we focus our attention on $4$-Engel groups. With the aid of the results of the first part we show that every $4$-Engel $3$-group is soluble and the derived length is bounded by some constant.
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Additional Information
  • Alireza Abdollahi
  • Affiliation: Department of Mathematics, University of Isfahan, Isfahan 81744, Iran
  • Email:
  • Gunnar Traustason
  • Affiliation: C.M.I.-Université de Provence, UMR-CNRS 6632, 39, rue F. Joliot-Curie, 13453 Marseille Cedex 13, France
  • Address at time of publication: Department of Mathematics, Lund Institute of Technology, P.O. Box 118, S-22100 Lund, Sweden
  • MR Author ID: 341715
  • Email:,
  • Received by editor(s): March 26, 2001
  • Received by editor(s) in revised form: May 12, 2001
  • Published electronically: March 12, 2002
  • Additional Notes: The second author thanks the European Community for their support with a Marie Curie grant.
  • Communicated by: Stephen D. Smith
  • © Copyright 2002 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 130 (2002), 2827-2836
  • MSC (2000): Primary 20F45, 20F50
  • DOI:
  • MathSciNet review: 1908264