2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/103625Let $G$ be a finite $p$-group, and let $\der Gd$ denote the $d$-th term of the derived series of $G$. We show, for $p\geq 5$, that $\der Gd\neq 1$ implies $\log_p\ord G\geq 2^d+3d-6$, and hence we improve a recent result by Mann.There are two related papers and version one had to be replaced after reogranisation of the material. Mathematical content is unchanged, but results have migrated from one paper to the otherGroup Theory20D15, 20-04text