Journal of Group Theory | 2021
Quasi-powerful π-groups
Abstract
Abstract In this paper, we introduce the notion of a quasi-powerful p-group for odd primes p. These are the finite p-groups G such that G/Z\u2062(G){G/Z(G)} is powerful in the sense of Lubotzky and Mann. We show that this large family of groups shares many of the same properties as powerful p-groups. For example, we show that they have a regular power structure, and we generalise a result of FernΓ‘ndez-Alcober on the order of commutators in powerful p-groups to this larger family of groups. We also obtain a bound on the number of generators of a subgroup of a quasi-powerful p-group, expressed in terms of the number of generators of the group, and we give an example which demonstrates this bound is close to best possible.