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Some conditions on infinite subsets of infinite groups. (English) Zbl 1006.20034

Summary: Let \(G\) be an infinite group. In this note we prove the following: For all \(a,b\in G\), \((ab)^2=(ba)^2\) if and only if every two infinite subsets \(X\) and \(Y\) of \(G\) contain elements \(x\) and \(y\), respectively, such that \((xy)^2=(yx)^2\). Also if \(n\in\{3,6\}\cup\{2^k\mid k\in\mathbb{N}\}\) then for all \(a,b\in G\), \(a^nb=ba^n\) if and only if every two infinite subsets \(X\) and \(Y\) of \(G\) contain elements \(x\) and \(y\), respectively, such that \(x^ny=yx^n\).

MSC:

20F99 Special aspects of infinite or finite groups
20F05 Generators, relations, and presentations of groups
20F12 Commutator calculus
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