Dimension subgroups and \(p\)-th powers in \(p\)-groups.
type: Item
instance of: scholarly article
title: Dimension subgroups and \(p\)-th powers in \(p\)-groups.
zbMATH Open document ID: 1046.20021
DOI: 10.1007/BF02783415
author: Lawrence E. Wilson
published in: Israel Journal of Mathematics
publication date: 2004-03-17T00:00:00Z
review text: \textit{C. M. Scoppola} and \textit{A. Shalev} [Isr. J. Math. 73, No. 1, 45-56 (1991; Zbl 0734.20008)] have proved that if the finite \(p\)-group \(G\) has nilpotence class strictly less than \(p^k-p^{k-1}\), then every product of \(p^k\)-th powers in \(G\) can be written as the \(p\)-th power of a suitable element of \(G\). They also showed that the result does not generally hold if the class of \(G\) is \(p^k\). In the paper under review, the author gives the best possible estimate, by showing that the result holds for groups of class strictly less than \(p^k\). The main tool is provided by the dimension subgroups, for which results of independent interest are proved.
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20F12
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 2060049
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: powers
zbMATH Keywords: commutators
zbMATH Keywords: dimension subgroups
reviewed by: Andrea Caranti
MaRDI profile type: Publication
cites work: The power-commutator structure of finite p-groups
cites work: Q3997907
cites work: On the structure of normal subgroups of potent \(p\)-groups.
cites work: A Contribution to the Theory of Groups of Prime-Power Order
cites work: Powerful \(p\)-groups. I: Finite groups.
cites work: The power structure of \(p\)-groups. I
cites work: Power closure and the Engel condition
cites work: Applications of dimension subgroups to the power structure of \(p\)-groups
cites work: Characterization of \(p\)-adic analytic groups in terms of wreath products
cites work: On the power structure of powerful p-groups
cites work: The power-commutator structure of certain finite p-groups
cites work: Q4660847
full work available at URL: https://doi.org/10.1007/bf02783415
OpenAlex ID: W2034735584
description: scientific article; zbMATH DE number 2060049