Finite \(p\)-groups with a fixed-point-free automorphisms of order seven.
type: Item
instance of: scholarly article
title: Finite \(p\)-groups with a fixed-point-free automorphisms of order seven.
zbMATH Open document ID: 1230.20022
author: Shousaku Abe
published in: Hokkaido Mathematical Journal
publication date: 2011-03-15T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.hokmj/1300108398
review text: By the Higman-Kreknin-Kostrikin theorem, if a nilpotent group \(G\) admits a fixed-point-free automorphism of prime order \(q\), then it is nilpotent of class \(\leq h(q)\). In the present paper, for \(q=7\), under the additional assumption that \(G\) is a finite \(p\)-group generated by two elements of order \(p\), it is proved that \(G\) is nilpotent of class less than 7. Some further information is obtained under the assumption that \(p\equiv 1\bmod 7\), which is always true if \(G\) is non-Abelian. It is known that \(h(7)=12\); thus a better bound is obtained due to the restriction on the number of generators. Reviewer's remarks: The author's avoidance of using Lie rings makes the proof unnecessarily long and cumbersome; plus, by using Lie rings the result can be extended to any two-generator \(p\)-group with a fixed-point-free automorphism of order 7.
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 5866038
zbMATH Keywords: fixed-point-free automorphisms
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: nilpotency classes
reviewed by: Evgenii I. Khukhro
MaRDI profile type: Publication
OpenAlex ID: W2080560325
DOI: 10.14492/HOKMJ/1300108398
description: scientific article; zbMATH DE number 5866038