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Groups which satisfy a Thue–Morse identity

H. Khosravi

Faculty of Basic Sciences and Engineering, Gonbad Kavous University, Gonbad, Iran

E-mail Address: Khosravi@gonbad.ac.ir

Corresponding author.

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and

A. Faramarzi Salles

School of Mathematics and Computer Science, Damghan University, Damghan, Iran

E-mail Address: faramarzi@du.ac.ir

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https://doi.org/10.1142/S0219498820501911Cited by:0 (Source: Crossref)

Abstract

In this paper, we study 3-Thue–Morse groups, but these are the groups satisfying the semigroup identity $x y^{2} x y x^{2} y = y x^{2} y x y^{2} x$ $x y^{2} x y x^{2} y = y x^{2} y x y^{2} x$ . We prove that if $G$ $G$ is a 3-Thue–Morse group then $⟨ x, x^{y} ⟩$ $〈 x, x^{y} 〉$ is soluble for every $x$ $x$ and $y$ $y$ in $G$ $G$ . Furthermore, if $G$ $G$ is an Engel group without involution then we show that $G$ $G$ is locally nilpotent.

Communicated by M. L. Lewis

Keywords:

AMSC: 20E25

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