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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

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.

MSC:

20F16 Solvable groups, supersolvable groups
20F12 Commutator calculus
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