14.43 (1999)

Suppose that a finite group $G$ has the form $G = AB$, where $A$ and $B$ are nilpotent subgroups of classes $\alpha$ and $\beta$ respectively; then $G$ is soluble (O. H. Kegel, H. Wielandt, 1961). Although the derived length $\operatorname{dl}(G)$ of $G$ need not be bounded by $\alpha + \beta$ (see 5.17), can one bound $\operatorname{dl}(G)$ by a (linear) function of $\alpha$ and $\beta$?