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 $G/Z(G)$ 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. We give an infinite family of examples which demonstrates this bound is close to best possible.

Cited by in corpus (1)

• On the regular power structure of $p$-groups and applications