8.76 (1982)

Give a realistic upper bound for the torsion-free rank of a finitely generated nilpotent group in terms of the ranks of its abelian subgroups. More precisely, for each integer $n$ let $f(n)$ be the largest integer $h$ such that there is a finitely generated nilpotent group of torsion-free rank $h$ with the property that all abelian subgroups have torsion-free rank at most $n$. It is easy to see that $f(n)$ is bounded above by $n(n+1)/2$. Describe the behavior of $f(n)$ for large $n$. Is $f(n)$ bounded below by a quadratic in $n$?