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Groups in which certain equations have many solutions. (English) Zbl 1072.20035
For a group word \(w(x_1,\dots,x_n)\) and a cardinal \(\kappa\), let \({\mathcal V}_\kappa(w)\) be the class of all groups \(G\) such that every subset of \(G\) of cardinality \(\kappa\) contains \(n\) distinct elements \(g_1,\dots,g_n\) with \(w(g_1,\dots,g_n)=1\). For example, if \(w\) is \([x_1,x_2]\) then \({\mathcal V}_\omega(w)\) is the class of all groups \(G\) without infinite sets of pairwise non-commuting elements; as B. H. Neumann proved [J. Aust. Math. Soc., Ser. A 21, 467-472 (1976; Zbl 0333.05110)] they are exactly the central-by-finite groups.
The author considers the word \(w=x_1^{\alpha_1}\cdots x_n^{\alpha_n}\), where \(\alpha_1,\dots,\alpha_n\) are nonzero integers. The main result: for an infinite group \(G\), the following are equivalent: (1) \(G\in{\mathcal V}_\omega(w)\); (2) \(G\in{\mathcal V}_m(w)\), for some \(m\in\omega\); (3) \(G\in{\mathcal B}_{\alpha_1+\cdots+\alpha_n}\cap\mathcal{FB}_\alpha\), where \(\alpha\) is the greatest common divisor of \(\alpha_1,\dots,\alpha_n\). Here \({\mathcal B}_\alpha\) is the variety of groups satisfying \(x^\alpha=1\), and \(\mathcal F\) is the class of finite groups.

MSC:
20E10 Quasivarieties and varieties of groups
20F05 Generators, relations, and presentations of groups
20F12 Commutator calculus
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References:
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