20.106 (2022)

Let $G$ be a residually finite 2-group and let $x \in G$ be a left 3-Engel element of order 2. Is $\langle x^G \rangle$ locally nilpotent?

It is known that in any group a 3-Engel element of odd order belongs to the locally nilpotent radical (E. Jabara, G. Traustason, Proc. Amer. Math. Soc., 147, no. 5 (2019), 1921–1927).

Comment of 2025: An element $a \in G$ is called a strong left 3-Engel element if $\langle a, a^g \rangle$ is nilpotent of class at most 2 and $\langle a, a^g, a^h \rangle$ is nilpotent of class at most 3 for all $g, h \in G$. (This is equivalent to $a$ being left 3-Engel when $a$ is of odd order.) It is proved that if $a$ is a strong left 3-Engel element in an arbitrary group $G$, then $\langle a \rangle^G$ is locally nilpotent (A. Hadjievangelou, G. Traustason, Proc. Amer. Math. Soc., 152, no. 4 (2024), 1467–1477).