Quasi-powerful 𝑝-groups

Author:

Williams James1

Affiliation:

1. School of Mathematics , University of Bristol , Bristol BS8 1UG , United Kingdom

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 ( 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.

Publisher

Walter de Gruyter GmbH

Subject

Algebra and Number Theory

Reference26 articles.

1. A. Abdollahi and G. Traustason, On locally finite 𝑝-groups satisfying an Engel condition, Proc. Amer. Math. Soc. 130 (2002), no. 10, 2827–2836.

2. J. D. Dixon, M. P. F. du Sautoy, A. Mann and D. Segal, Analytic Pro-𝑝 Groups, 2nd ed., Cambridge Stud. Adv. Math. 61, Cambridge University, Cambridge, 1999.

3. G. A. Fernández-Alcober, Omega subgroups of powerful 𝑝-groups, Israel J. Math. 162 (2007), 75–79.

4. G. A. Fernández-Alcober and I. de las Heras, Commutators in finite 𝑝-groups with 2-generator derived subgroup, Israel J. Math. 232 (2019), no. 1, 109–124.

5. J. González-Sánchez and A. Jaikin-Zapirain, On the structure of normal subgroups of potent 𝑝-groups, J. Algebra 276 (2004), no. 1, 193–209.

Cited by 2 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. The powerful class of groups;Mathematische Nachrichten;2023-10-30

2. The Wreath Product of Powerful p-Groups;Symmetry;2023-10-27

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