Small derived quotients in finite p-groups

Original · EN

More than 70 years ago, P. Hall showed that if G is a finite p-group such that a term Gd+1 of the derived series is non-trivial, then the order of the quotient Gd/ Gd+1 is at least p²ᵈ⁺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 p⁶.