Groups with regular elementary 2-groups of automorphisms
type: Item
instance of: scholarly article
title: Groups with regular elementary 2-groups of automorphisms
zbMATH Open document ID: 0739.20017
DOI: 10.1007/BF01980740
author: Pavel Shumyatsky
published in: Algebra and Logic
publication date: 1992-06-25T00:00:00Z
review text: Theorem 1. There exists a function \(f(x,y)\) of two natural variables such that any \(k\)-step solvable, periodic group \(G\) admitting a regular elementary group of automorphisms of order \(2^ n\) has an invariant series \(G=H_ 1\supseteq H_ 2\supseteq\cdots\supseteq H_{n+1}=1\), in which the factors are nilpotent and the nilpotent length of \(H_ i/H_{i+1}\) is at most \(f(i,k)\), \(1\leq i\leq n\). Theorem 2. Suppose \(G\) is a periodic group admitting a regular elementary group of automorphisms of order \(2^ n\). If some term of the derived series of \(G\) having a natural subscript is hypercentral, then \(G\) has an invariant series \(G=H_ 1\supseteq H_ 2\supseteq\dots\supseteq H_{n+1}=1\) in which all of the factors are hypercentral.
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20F14
Mathematics Subject Classification ID: 20F28
Mathematics Subject Classification ID: 20F16
zbMATH DE Number: 3974
zbMATH Keywords: solvable periodic groups
zbMATH Keywords: regular elementary groups of automorphisms
zbMATH Keywords: invariant series
zbMATH Keywords: nilpotent lengths
zbMATH Keywords: derived series
zbMATH Keywords: hypercentral groups
MaRDI profile type: Publication
cites work: Q5330689
cites work: The Klein group as an automorphism group without fixed point
cites work: On groups admitting fixed point free abelian operator groups
cites work: Fitting height of groups and of fixed points
description: scientific article; zbMATH DE number 3974