Quasi-powerful 𝑝-groups
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 ( 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 Fernández-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.