On the nilpotency class of a certain subgroup

Question

Let $G$ be a nilpotent group of nilpotency class $c>2$ and $a\in G‎\setminus G'$.

Is the nilpotency class of $\langle a\rangle G'$ less than c?

Answer 1

Yes for $c\ge 2$ (rather than $c>2$). By induction on $c$: trivial for $c=2$. For $c\ge 3$ larger, let $Z$ be the last nontrivial term of the lower central series, so $G/Z$ has class equal to $c-1$ and $Z\subset G'$. By induction, $\langle a\rangle G'/Z$ has class $<c-1$. Since $Z$ is central in $\langle a\rangle G'$, this implies that $\langle a\rangle G'$ has class $\le c-1$.