15.61 (2002)

Is it true that $l_n^\pi(G) \leqslant n(G_\pi) - 1 + \max_{p \in \pi} l_p(G)$ for any $\pi$-soluble group $G$? Here $n(G_\pi)$ is the nilpotent length of a Hall $\pi$-subgroup $G_\pi$ of the group $G$ and $l_n^\pi(G)$ is the nilpotent $\pi$-length of $G$, that is, the minimum number of $\pi$-factors in those normal series of $G$ whose factors are either $\pi'$-groups, or nilpotent $\pi$-groups. The answer is known to be affirmative in the case when all proper subgroups of $G_\pi$ are supersoluble.