Sandwich groups and (strong) left 3-Engel elements in groups

Hadjievangelou, Anastasia; Traustason, Gunnar

doi:10.1090/proc/16695

Sandwich groups and (strong) left $3$-Engel elements in groups
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by Anastasia Hadjievangelou and Gunnar Traustason;

Proc. Amer. Math. Soc. 152 (2024), 1467-1477

DOI: https://doi.org/10.1090/proc/16695

Published electronically: February 9, 2024

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Abstract:

In this paper we prove a group theoretic analogue of the well known local nilpotence theorem for sandwich Lie algebras due to Kostrikin and Zel’manov [Trudy Mat. Inst. Steklov. 183 (1990), pp. 106–111, 225]. We introduce the notion of a strong left $3$-Engel element of a group $G$ and show that these are always in the locally nilpotent radical of $G$. This generalises a previous result of Jabara and Traustason [Proc. Amer. Math. Soc. 147 (2019), pp. 1921–1927] that showed that a left $3$-Engel element $a$ of a group $G$ is in the locally nilpotent radical of $G$ whenever $a$ is of odd order.

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