Solvability and maximal cyclic subgroups

For a finite group [Formula: see text], let [Formula: see text] denote the smallest index of a maximal cyclic subgroup of [Formula: see text], with [Formula: see text] if [Formula: see text] is cyclic. We prove that [Formula: see text] implies that [Formula: see text] is solvable, and that this bound is sharp. We also obtain bounds for the derived length of finite solvable groups in terms of [Formula: see text], improving earlier estimates based on the number [Formula: see text] of maximal cyclic subgroups. In particular, for finite [Formula: see text]-groups, we obtain a bound of order [Formula: see text].