Abdollahi, Alireza; Trabelsi, Nadir Some extensions of a problem of Paul Erdős on groups. (Quelques extensions d’un problème de Paul Erdős sur les groupes.) (French) Zbl 1041.20022 Bull. Belg. Math. Soc. - Simon Stevin 9, No. 2, 205-215 (2002). A group \(G\) is said to have ‘the property \(E^*\)’ if, for every infinite subset \(X\) of \(G\), there exist distinct elements \(x,y\in X\), a positive integer \(n\) and non-zero integers \(t_0,t_1,\dots,t_n\) such that \([z_0^{t_0},z_1^{t_1},\dots,z_n^{t_n}]=1\) for \(z_i\) in \(\{x,y\}\). If all the \(t_i\) can be chosen to be \(\pm 1\), then \(G\) has ‘the property \(E^\#\)’. The authors study finitely generated soluble groups with these properties, thus continuing a line of research begun by P. Erdős and B. H. Neumann. We quote two of the many results obtained. Theorem 1.1. A finitely generated soluble group has \(E^*\) if and only if it is nilpotent-by-finite. Theorem 1.4. A finitely generated soluble group has \(E^\#\) if and only if it is finite-by-nilpotent. Reviewer: Derek J. S. Robinson (Urbana) Cited in 2 ReviewsCited in 3 Documents MSC: 20F16 Solvable groups, supersolvable groups 20F12 Commutator calculus Keywords:commutators; infinite subsets PDFBibTeX XMLCite \textit{A. Abdollahi} and \textit{N. Trabelsi}, Bull. Belg. Math. Soc. - Simon Stevin 9, No. 2, 205--215 (2002; Zbl 1041.20022)