Can one find a function $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for every finite supersolvable group $G$ we have: $d(G') \leq f(d(G))$?


Here $d(K)$ is the cardinality of a minimal set of generators for $K$, and $K'$ is the commutator subgroup of $K$.



Number of generators of the commutator
By a method similar to Jeremy Rouse's comment on my original (incorrect) answer, I think the answer is no. A $p$-group is certainly supersolvable. If we take $G = C\_{p} \wr C\_{p^{n}},$ then $d(G) =2,$ $d(G^{\prime}) = p^{n}-1$ - in fact, the case $n = 1$ already seems to provide a counterexample to the question as asked, since $p$ is an arbitrary prime, but this shows that there is no bound in general, even if we allow a function which depends on the prime divisors as well as $d(G).$


