Given a finite nilpotent group $G$ and let us denote by $L_n(G)$ the set of left $n$Engel elements in $G$.
Assume that $n$ is the smallest positive integer such that $L_n(G)=G$.
Is it true that $G$ is always generated by $GL_{n1}(G)$?
Given a finite nilpotent group $G$ and let us denote by $L_n(G)$ the set of left $n$Engel elements in $G$. Assume that $n$ is the smallest positive integer such that $L_n(G)=G$. Is it true that $G$ is always generated by $GL_{n1}(G)$? 

