A combinatorial condition on a certain variety of groups (Q5956892)

Let \(w(x_1,\dots,x_n)\) be a reduced word in the free group \(F\) freely generated by the countable set \(\{x_1,x_2,\dots\}\) and let \(\nu\) be the variety of groups defined by the law \(w(x_1,\dots,x_n)\). Let also \(\nu^*\) be the class of all groups such that \(G\) is in \(\nu^*\), if and only if for all infinite subsets \(X_1,\dots,X_n\) of \(G\), there exist elements \(g_i\in X_i\), \(i=1,\dots,n\), such that \(w(g_1,\dots,g_n)=1\). \textit{P. S. Kim, A. H. Rhemtulla} and \textit{H. Smith} [Houston J. Math. 17, No. 3, 429-437 (1991; Zbl 0744.20033)] raised the following question that whether, for every word \(w(x_1,\dots,x_n)\), \(\nu^*=\nu\cup{\mathcal F}\), where \(\mathcal F\) is the class of finite groups. Many authors have shown that the answer is positive, for certain words. See for instance [\textit{L. S. Spiezia}, Arch. Math. 64, No. 5, 369-373 (1995; Zbl 0823.20038)]. In the paper under review, the author proves that the question has positive answer for the variety \({\mathcal B}_n\) (the so called variety of \(n\)-Bell groups) defined by the law \(w(x_1,x_2)=[x_1^n,x_2][x_1,x_2^n]^{-1}\), for \(n=2,-2\) or \(3\) (Theorem A). It is also shown that, if an infinite group \(G\) has finitely many elements of order 2 or 3, then \(G\in{\mathcal B}_n^*\) if and only if \(G\in{\mathcal B}_n\), for \(n=-3\) or \(4\) (Theorem B). To prove his results, the author uses some properties of Engel groups and results of \textit{G. Endimioni} [Commun. Algebra 23, No. 14, 5297-5307 (1995; Zbl 0859.20021)]. Reviewer's remark: For a group \(G\) and integer \(n\), the set \(B(G,n)=\{x\in G:[x^n,y]=[y,x^n]\), for all \(y\in G\}\) is called the \(n\)-Bell centre of \(G\). The author claims that; whether the \(n\)-Bell centre of a group always forms a subgroup is an open problem! One should note that \textit{M. L. Newell} [Infinite groups 94. Proceedings of the international Conference, Ravello, May 23-27, 1994. Berlin, Walter de Gruyter 205-213 (1996; Zbl 0871.20033)] has verified that for a certain group, \(B(G,4)\) does not form a subgroup. There are also some misprints in different places of the paper; for instace: on page 457, the commutator identity \([x,y^{-1}]=[x,z][x,y]^{-y^{-1}}\) should be written as \([x,y^{-1}]=[x,y]^{-y^{-1}}\), and the last line on the same page \((x_{i\sigma(1)})^{x_{i\sigma(2)}}\) must be \((x_{i\sigma(1)}^n)^{x_{i\sigma(2)}}\).