Groups with identities
type: Item
instance of: scholarly article
title: Groups with identities
zbMATH Open document ID: 0697.20013
DOI: 10.1016/0168-0072(89)90060-2
author: Françoise Point
published in: Annals of Pure and Applied Logic
publication date: 1989-01-01T00:00:00Z
review text: This paper is concerned with groups which satisfy a finite ``monoidal disjunction'' \(\bigvee^{n}_{i=1}w_ i(x,y)=v_ i(x,y)\) where, for each i, \(v_ i\) and \(w_ i\) are distinct elements of the free monoid on x and y. It was shown by \textit{M. Boffa} [C. R. Acad. Sci., Paris, Sér. I 303, 587-589 (1986; Zbl 0619.03026)] that these groups may also be described as those admitting the elimination of inverses (e.i.). The author defines what it means for a group to admit e.i. with complexity \(\leq \ell\) (where \(\ell\) is a positive integer). It is immediate from the definition that every group which admits e.i. must admit e.i. with complexity \(\leq \ell\) for some \(\ell\). One main result of the paper is that if G is a soluble group which admits e.i. then G is uniformly locally quasi-nilpotent: in other words, for every k there exist c and i such that every k-generator subgroup of G has a nilpotent subgroup of class c and of finite index i. Moreover, if G admits e.i. with complexity \(\leq \ell\) then c can be computed explicitly in terms of k, \(\ell\) and the solubility length of G. Another main result (using the classification of finite simple groups) is that if G is a locally finite group which admits e.i. then G has a locally nilpotent normal subgroup N such that G/N has finite exponent.
Mathematics Subject Classification ID: 20E10
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 03C05
Mathematics Subject Classification ID: 20E25
Mathematics Subject Classification ID: 20F50
zbMATH DE Number: 4142296
zbMATH Keywords: monoidal disjunction
zbMATH Keywords: elimination of inverses
zbMATH Keywords: soluble group
zbMATH Keywords: locally quasi-nilpotent
zbMATH Keywords: k-generator subgroup
zbMATH Keywords: nilpotent subgroup
zbMATH Keywords: solubility length
zbMATH Keywords: locally finite group
zbMATH Keywords: locally nilpotent normal subgroup
reviewed by: Roger M. Bryant
MaRDI profile type: Publication
full work available at URL: 0168-0072(89)90060-2
OpenAlex ID: W2073206611
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description: scientific article; zbMATH DE number 4142296