2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/103627More than 70 years ago, P. Hall showed that if $G$ is a finite $p$-group such that a term $\der G{d+1}$ of the derived series is non-trivial, then the order of the quotient $\der Gd/\der G{d+1}$ is at least $p^{2^d+1}$. Recently Mann proved that, in a finite $p$-group, Hall's lower bound can be taken for at most two distinct $d$. We improve this result and show that if $p$ is odd, then it can only be taken for two distinct $d$ in a group with order $p^6$.Two related papers have been submitted. The material have been reorganised for Versions 2 and results migrated between papersGroup Theory20D15, 20-04text