A condition on a certain variety of groups
type: Item
instance of: scholarly article
title: A condition on a certain variety of groups
zbMATH Open document ID: 1013.20021
author: Alireza Abdollahi
author: Bijan Taeri
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 2003-03-18T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108531
review text: Let \(\mathcal V\) be a variety of groups defined by the law \(v(x_1,\dots,x_n)=1\). A group \(G\) is said to be a \({\mathcal V}^*\)-group if for every set of \(n\) infinite subsets \(X_1,\dots,X_n\) of \(G\), there are elements \(g_i\in X_i\) such that \(v(g_1,\dots,g_n)=1\). \textit{P. S. Kim, A. Rhemtulla} and \textit{H. Smith} [Houston J. Math. 17, No. 3, 429-437 (1991; Zbl 0744.20033)] asked for which varieties \(\mathcal V\) every infinite \({\mathcal V}^*\)-group is in \(\mathcal V\). Positive answers are known for a number of varieties, including the variety of all Abelian groups, varieties of nilpotent groups of bounded class, and, of particular relevance to this paper, the Burnside variety, \({\mathcal B}_d\), of groups of exponent \(d\). A related question, posed by \textit{C. Delizia} [Algebra Colloq. 2, No. 2, 97-104 (1995; Zbl 0839.20047)], asks whether, given varieties \(\mathcal V\) and \(\mathcal W\) defined by the laws \(v(x_1,\dots,x_n)=1\) and \(w(x_1,\dots,x_l)=1\), respectively, \(\mathcal V=\mathcal W\) implies \({\mathcal V}^*={\mathcal W}^*\). In this paper it is shown that, if \(\mathcal V\) is the variety defined by the law \(x_1^{a_1}\cdots x_n^{a_n}=1\), where \(a_1,\dots,a_n\) are nonzero integers with greatest common divisor \(d\), then \({\mathcal V}^*={\mathcal B}_d^*\).
reviewed by: Sheila Oates-Williams
Mathematics Subject Classification ID: 20E10
zbMATH DE Number: 1884329
zbMATH Keywords: varieties of groups
zbMATH Keywords: exponents
zbMATH Keywords: Burnside varieties
zbMATH Keywords: combinatorial conditions on infinite subsets
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1884329