17.4 (2010)

Let $x$ be a right 4-Engel element of a group $G$.
$\qquad$ a) Is it true that the normal closure $\langle x \rangle^G$ of $x$ in $G$ is nilpotent if $G$ is locally nilpotent?
$\qquad$ b) If the answer to a) is affirmative, is there a bound on the nilpotency class of $\langle x \rangle^G$?
$\qquad$ c) Is it true that $\langle x \rangle^G$ is always nilpotent?