On numbers which are orders of nilpotent groups with bounded class
type: Item
instance of: scholarly article
title: On numbers which are orders of nilpotent groups with bounded class
author: Maria Ferrara
published in: International Journal of Group Theory
publication date: 2024-10-16T00:00:00Z
review text: Let \(m \geq 2\) be an integer and let \(m=p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}} \ldots p_{r}^{\alpha_{r}}\) be its prime factorization. A result by \textit{T. W. Müller} [J. Algebra 300, No. 1, 10--15 (2006; Zbl 1099.20014)] states that every group of order \(m\) is nilpotent of class at most \(n\) if and only if (a) \(p_{i}\) does not divide \(p_{j}^{k} -1\), for all \(i \not =j\) and all \(k \in [1,\dots, \alpha_{j}]\), (b) \(1 \leq \alpha_i \leq n+1\), for all \(i\).\N\NThe author characterizes those numbers \(m\) for which any group of order \(m\) is an \(n\)-Engel group and those numbers \(m\) for which any group of order \(m\) has all its subgroups subnormal of defect at most \(n\) (they are essentially the same that satisfy conditions (a) and (b) described above).
reviewed by: Egle Bettio
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20F45
Mathematics Subject Classification ID: 20E15
zbMATH DE Number: 7927532
zbMATH Keywords: Engel group
zbMATH Keywords: subnormal subgroup
zbMATH Keywords: defect
MaRDI profile type: Publication
DOI: 10.22108/IJGT.2023.137950.1851
description: scientific article; zbMATH DE number 7927532