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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
##### Keywords:
varieties of groups; group words
Full Text:
##### References:
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