A condition on a certain variety of groups
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 \({\mathcal V}^*={\mathcal B}_d^*\).
combinatorial conditions on infinite subsets
scientific article; zbMATH DE number 1884329