On the rank of a verbal subgroup of a finite group
We show that if $w$ is a multilinear commutator word and $G$ a finite group in which every metanilpotent subgroup generated by $w$-values is of rank at most $r$, then the rank of the verbal subgroup $w(G)$ is bounded in terms of $r$ and $w$ only. In the case where $G$ is soluble we obtain a better result -- if $G$ is a finite soluble group in which every nilpotent subgroup generated by $w$-values is of rank at most $r$, then the rank of $w(G)$ is at most $r+1$.