Left $3$-Engel elements of odd order in groups
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- by Enrico Jabara and Gunnar Traustason PDF
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Abstract:
Let $G$ be a group and let $x\in G$ be a left $3$-Engel element of odd order. We show that $x$ is in the locally nilpotent radical of $G$. We establish this by proving that any finitely generated sandwich group, generated by elements of odd orders, is nilpotent. This can be seen as a group theoretic analog of a well-known theorem on sandwich algebras by Kostrikin and Zel’manov.
We also give some applications of our main result. In particular, for any given word $w=w(x_{1},\ldots ,x_{n})$ in $n$ variables, we show that if the variety of groups satisfying the law $w^{3}=1$ is a locally finite variety of groups of exponent $9$, then the same is true for the variety of groups satisfying the law $(x_{n+1}^{3}w^{3})^{3}=1$.
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Additional Information
- Enrico Jabara
- Affiliation: Dipartimento di Filosofia e Beni Culturali, Università Ca’Foscari Venezia, Dorsoduro, 3246, 30123 Venezia VE, Italy
- MR Author ID: 202331
- Gunnar Traustason
- Affiliation: Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom
- MR Author ID: 341715
- Received by editor(s): November 29, 2017
- Received by editor(s) in revised form: August 31, 2018
- Published electronically: January 18, 2019
- Additional Notes: The second author was supported through a standard grant from EPSRC
- Communicated by: Pham Huu Tiep
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 1921-1927
- MSC (2010): Primary 20F05, 20F45
- DOI: https://doi.org/10.1090/proc/14389
- MathSciNet review: 3937670