Quasi-powerful $p$-groups
James Williams

TL;DR
This paper introduces quasi-powerful p-groups, a new class of finite p-groups where the quotient by the center is powerful, sharing many properties with powerful p-groups and extending known results.
Contribution
It defines quasi-powerful p-groups, explores their properties, and generalizes existing results on powerful p-groups to this broader class.
Findings
Quasi-powerful p-groups have a regular power structure.
A bound on the number of generators of subgroups is established.
An infinite family of examples demonstrates the bound's near optimality.
Abstract
In this paper we introduce the notion of a quasi-powerful -group for odd primes . These are the finite -groups such that 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 -groups. For example, we show that they have a regular power structure, and we generalise a result of Fern\'andez-Alcober on the order of commutators in powerful -groups to this larger family of groups. We also obtain a bound on the number of generators of a subgroup of a quasi-powerful -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.
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Taxonomy
TopicsFinite Group Theory Research · Coding theory and cryptography · graph theory and CDMA systems
