Small derived quotients in finite p-groups

Abstract

More 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 $p^6$.Comment: Two related papers have been submitted. The material have been reorganised for Versions 2 and results migrated between paper