Abdollahi, Alireza; Taeri, Bijan Some conditions on infinite subsets of infinite groups. (English) Zbl 1006.20034 Bull. Malays. Math. Soc., II. Ser. 22, No. 1, 87-93 (1999). 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 \(x^ny=yx^n\). Cited in 1 ReviewCited in 2 Documents MSC: 20F99 Special aspects of infinite or finite groups 20F05 Generators, relations, and presentations of groups 20F12 Commutator calculus Keywords:combinatorial conditions on infinite subsets; infinite groups PDFBibTeX XMLCite \textit{A. Abdollahi} and \textit{B. Taeri}, Bull. Malays. Math. Soc., II. Ser. 22, No. 1, 87--93 (1999; Zbl 1006.20034) Full Text: EuDML