Finiteness conditions for a group of finite exponent
type: Item
instance of: scholarly article
title: Finiteness conditions for a group of finite exponent
zbMATH Open document ID: 0797.20032
author: Gérard Endimioni
published in: Journal of Algebra
publication date: 1993-05-16T00:00:00Z
review text: A periodic group \(G\) all of whose \(d\)-generator subgroups (for fixed \(d > 1\)) are solvable need not itself be solvable (Golod), and similarly for nilpotence. However if \(G\) is finite, then the solvability of every 2- generator subgroup entails the solvability of \(G\) (Thompson), and likewise for nilpotence. By strengthening the hypothesis the author obtains in the present paper similar results without the finiteness assumption. Let \(d\), \(d'\) (\(d \geq 1\), \(d' \geq 0\)) be integers. A group is called \((d,d')\)-solvable if each \(d\)-generator subgroup is solvable of length \(\leq d'\). Theorem 1. A periodic \((d,d-1)\)-solvable group (for some \(d \geq 1\)) is locally finite (and so certainly locally solvable). The analogous result for a \((d,2^ d-2)\)-nilpotent group (for some \(d \geq 1\)) also holds. An application is given to the question of \textit{Hanna Neumann} [from her book ``Varieties of Groups'' (1967; Zbl 0251.20001)] as to whether for each integer \(e > 0\) there exists an integer \(d > 0\) such that for a group of exponent \(e\) to be locally finite it suffices that the \(d\)- generator subgroups be finite. It is shown that for \(e \leq 6\) or \(e = 10\), 12, 15 or 20, the answer is in the affirmative. Finally a result on \((d,2^ d-2)\)-nilpotent groups of finite exponent is proved using the result of Zelmanov on the nilpotence of Lie algebras over a field of characteristic zero satisfying an Engel condition.
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20F19
Mathematics Subject Classification ID: 20E25
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20E10
zbMATH DE Number: 169078
zbMATH Keywords: finiteness conditions
zbMATH Keywords: periodic group
zbMATH Keywords: \(d\)-generator subgroups
zbMATH Keywords: solvability
zbMATH Keywords: nilpotence
zbMATH Keywords: \((d,d')\)-solvable
zbMATH Keywords: locally finite
zbMATH Keywords: locally solvable
zbMATH Keywords: \((d,2^ d-2)\)-nilpotent groups of finite exponent
MaRDI profile type: Publication
full work available at URL: jabr.1993.1045
OpenAlex ID: W2035700684
DOI: 10.1006/JABR.1993.1045
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description: scientific article; zbMATH DE number 169078