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A finiteness condition concerning commutators in groups. (English) Zbl 0813.20026

In a paper by P. S. Kim, A. H. Rhemtulla and H. Smith [Houston J. Math. 17, 429-437 (1991; Zbl 0744.20033)], the following question was asked. Let \(V\) be a variety of groups defined by means of a law \(v(x_ 1,\dots,x_ n) = 1\). Let \(G\) be an infinite group with the property that for all choices of infinite subsets \(X_ 1,\dots,X_ n\) of \(G\) there exist elements \(x_ 1,\dots,x_ n\), with \(x_ i \in X_ i\) for all \(i\), such that \(v(x_ 1,\dots,x_ n) = 1\). Is it true that \(G \in V\)? In the paper under review the authors prove that the answer is yes for the variety defined by the law \([x,y]^ 2 = 1\). This is one of the simplest cases where the answer was previously unknown.

MSC:

20E10 Quasivarieties and varieties of groups
20F12 Commutator calculus

Citations:

Zbl 0744.20033
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