When right n-Engel elements of a group form a subgroup?

Abstract

Let $R_n(G)$ denotes the set of all right $n$-Engel elements of a group $G$. We show that in any group $G$ whose 5th term of lower central series has no element of order 2, $R_3(G)$ is a subgroup. Furthermore we prove that $R_4(G)$ is a subgroup for locally nilpotent groups $G$ without elements of orders 2, 3 or 5; and in this case the normal closure $ ^G$ is nilpotent of class at most 7 for each $x\in R_4(G)$. Using a group constructed by Newman and Nickel we also show that, for each $n\geq 5$, there exists a nilpotent group of class $n+2$ containing a right $n$-Engel element $x$ and an element $a\in G$ such that both $[x^{-1},_n a]$ and $[x^{k},_n a]$ are of infinite order for all integers $k\geq 2$. We finish the paper by proving that at least one of the following happens: (1) There is an infinite finitely generated $k$-Engel group of exponent $n$ for some positive integer $k$ and some 2-power number $n$. (2) There is a group generated by finitely many bounded left Engel elements which is not an Engel group.