16.66 (2006)

For a group $G$, let $D_n(G)$ denote the $n$-th dimension subgroup of $G$, and $\zeta_n(G)$ the $n$-th term of its upper central series. For a given integer $n \geqslant 1$, let $f(n) = \max\{m \mid \exists$ a nilpotent group $G$ of class $n$ with $D_m(G) \neq 1\}$ and $g(n) = \max\{m \mid \exists$ a nilpotent group $G$ of class $n$ such that $D_n(G) \not\subseteq \zeta_m(G)\}$.
$\qquad$ a) What is $f(3)$?
$\qquad$ b) (B. I. Plotkin). Is it true that $f(n)$ is finite for all $n$?
$\qquad$ c) Is the growth of $f(n)$ and $g(n)$ polynomial, exponential, or intermediary?