Right Engel elements of stability groups of general series in vector spaces

Abstract

Let $V$ be an arbitrary vector space over some division ring $D$, $\mathbf{L}$ a general series of subspaces of $V$ covering all of $V\backslash \{0\}$ and $S$ the full stability subgroup of $\mathbf{L}$ in $\operatorname{GL}(V)$. We prove that always the set of bounded right Engel elements of $S$ is equal to the $\omega$-th term of the upper central series of $S$ and that the set of right Engel elements of $S$ is frequently equal to the hypercentre of $S$.