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On locally finite $p$-groups satisfying an Engel condition

Authors: Alireza Abdollahi and Gunnar Traustason
Journal: Proc. Amer. Math. Soc. 130 (2002), 2827-2836
MSC (2000): Primary 20F45, 20F50
Published electronically: March 12, 2002
MathSciNet review: 1908264
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Abstract: 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

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

Keywords: Locally finite $p$-groups, Engel groups
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
Article copyright: © Copyright 2002 American Mathematical Society