Finite p-groups all of whose d-generated subgroups are powerful

Abstract

Let p be an odd prime. We study d-powerful p-groups, i.e., finite p-groups all of whose d-generated subgroups are powerful. For p≥5, we show that any powerful p-group G is d(G)-powerful, where d(G) denotes the minimal number of generators of G. Moreover, for a finite p-group G we prove that if there is a positive integer 2≤m<d(G) such that G is m-powerful, then all subgroups of G are powerful, i.e., G is modular. Our results lead to a new characterization of dually strongly balanced finite p-groups. Furthermore, minimal non-powerful finite p-groups are determined completely, up to isomorphism.For finitely generated pro-p groups some analogous results are proved. For instance, if for a torsion-free finitely generated pro-p group G there is a positive integer 2≤m<d(G) such that all m-generated subgroups of G are powerful, then all subgroups of G are powerful.