Narrow Results

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