Left $3$-Engel elements of odd order in groups

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 $(x_{n+1}^{3}w^{3})^{3}=1$.