## 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

## 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.## References

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## Additional Information

**Alireza Abdollahi**- Affiliation: Department of Mathematics, University of Isfahan, Isfahan 81744, Iran
- Email: alireza_abdollahi@yahoo.com
**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: gunnar@gyptis.univ-mrs.fr, gt@maths.lth.se
- 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: https://doi.org/10.1090/S0002-9939-02-06421-3
- MathSciNet review: 1908264