A question of Paul Erdős and nilpotent-by-finite groups (Q2758187)

\textit{B. H. Neumann} in response to a question of Paul Erdős proved [in J. Aust. Math. Soc., Ser. A 21, 467-472 (1976; Zbl 0333.05110)] that a group is center-by-finite if and only if every infinite subset contains a commuting pair of distinct elements. Later extensions of Erdős' question with different aspects have been considered by many people [see e.g. \textit{J. C. Lennox, J. Wiegold}, J. Aust. Math. Soc., Ser. A 31, 459-463 (1981; Zbl 0492.20019) and \textit{A. Abdollahi, B. Taeri}, Commun. Algebra 27, No. 11, 5633-5638 (1999; Zbl 0942.20014)].NEWLINENEWLINENEWLINEHere the author considers some combinatorial conditions on infinite or finite subsets of groups as follows: Let \(n\) be a positive integer or infinity (denoted \(\infty\)) and \(k\) be a positive integer. It is denoted by \(\Omega_k(n)\) (respectively, \({\mathcal U}_k(n)\)) the class of groups \(G\) in which every subset \(X\) of cardinality \(n+1\) (if \(n=\infty\) then \(n+1=\infty\)) contains distinct elements \(x,y\) and there exist non-zero integers \(t_0,t_1,\dots,t_k\) such that \([x_0^{t_0},x_1^{t_1},\dots,x_k^{t_k}]=1\), where \(x_i\in\{x,y\}\), \(i=0,1,\dots,k\), \(x_0\not=x_1\) (respectively, \(x_i^{t_i}\not=1\) for all \(i\in\{0,1,\dots,k\}\)). If the non-zero integers \(t_0,t_1,\dots,t_k\) are the same for any subset \(X\) of \(G\), we say that \(G\) is in the class \(\overline\Omega_k(n)\) (respectively, \(\overline{\mathcal U}_k(n)\)). It is denoted by \({\mathcal W}^*_k\) the class of groups \(G\) such that in every two infinite subsets \(X\) and \(Y\) of \(G\) there exist \(x\in X\) and \(y\in Y\) and there exist non-zero integers \(t_2,\dots,t_k\) such that \([x_0,x_1,x_2^{t_2},\dots,x_k^{t_k}]=1\), where \(x_i\in\{x,y\}\) for \(i=0,1,\dots,k\) and \(x_0\not=x_1\). If the non-zero integers \(t_2,\dots,t_k\) are the same for any two infinite subsets \(X\) and \(Y\) of \(G\), we say that \(G\) is in \(\overline{\mathcal W}^*_k\). It is denoted by \({\mathcal E}_k(n)\) (respectively, \({\mathcal E}(n)\)) the class of all groups \(G\) in which every subset of cardinality \(n+1\) contains two distinct elements \(x,y\) such that \([x,{_ky}]=1\) (respectively, \([x,{_ty}]=1\) for some positive integer \(t\) depending on \(x,y\)). Finally it is denoted by \({\mathcal E}^*_k\) the class of all groups \(G\) in which every two infinite subsets \(X\) and \(Y\) contain elements \(x\in X\) and \(y\in Y\) such that \([x,{_ky}]=1\). (The author in his definitions has not indicated that the \(t_i\)'s must be non-zero, but he uses this fact in the proofs.) Obviously we have NEWLINE\[NEWLINE{\mathcal E}_k(n)\subseteq\overline\Omega_k(n)\subseteq\Omega_k(n)\subseteq\Omega_k(n+1)\subseteq\Omega_k(\infty)NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\overline{\mathcal U}_k(n)\subseteq{\mathcal U}_k(n)\subseteq\Omega_k(n)\text{ and }{\mathcal E}_k^*\subseteq\overline{\mathcal W}^*_k\subseteq{\mathcal W}^*_k\subseteq\Omega_k(\infty).NEWLINE\]NEWLINE The author in Theorem 2 of the paper under review proves that every finitely generated soluble group in \(\Omega_k(\infty)\) is nilpotent-by-finite. This result generalizes a result of \textit{N. Trabelsi} [in Bull. Aust. Math. Soc. 61, No. 1, 33-38 (2000; Zbl 0959.20034)] where a similar combinatorial condition on any two infinite subsets was considered. The proof of Theorem 2 depends on a deep result of \textit{P. H. Kropholler} [Proc. Lond. Math. Soc., III. Ser. 49, 155-169 (1984; Zbl 0537.20013)]. \textit{P. Longobardi} and \textit{M. Maj} [in Rend. Semin. Mat. Univ. Padova 89, 97-102 (1993; Zbl 0797.20031)] proved that a finitely generated soluble group \(G\in{\mathcal E}(\infty)\) if and only if \(G\) is finite-by-nilpotent. The reviewer has proved that a finitely generated residually finite \({\mathcal E}_k(n)\)-group, \(n\) a positive integer, is finite-by-nilpotent. The author proves in Theorem 3 of this paper that every finitely generated residually finite group in \(\overline\Omega_k(n)\) (\(n\) a positive integer) is nilpotent-by-finite. The proof of Theorem 3 depends on a result of \textit{J. S. Wilson} [Bull. Lond. Math. Soc. 23, No. 3, 239-248 (1991; Zbl 0746.20018)]. \textit{O. Puglisi} and \textit{L. S. Spiezia} proved [in Commun. Algebra 22, No. 4, 1457-1465 (1994; Zbl 0803.20024)] that every infinite locally finite or locally soluble \({\mathcal E}_k^*\)-group is a \(k\)-Engel group. The reviewer [in Bull. Aust. Math. Soc. 62, No. 1, 141-148 (2000; Zbl 0964.20019)] improved the latter result for locally graded groups (recall that a group is called locally graded if every nontrivial finitely generated subgroup has a nontrivial finite quotient). Here the author proves in Theorem 4 that every locally graded group in \(\overline{\mathcal W}^*_k\) is nilpotent-by-finite.NEWLINENEWLINENEWLINEReviewer's remark: The proof of Theorem 1 of the paper which says that every finite group in the class \({\mathcal U}_k(2)\) is nilpotent, relies on the \textit{false} assumption that the class \({\mathcal U}_k(2)\) is closed under taking quotients; in fact one can find a counter-example to show that Theorem 1 is false. In the proof of Corollary 3 of the paper, the author has used the consequence of Theorem 1 that every finite \(\overline{\mathcal U}_k(2)\)-group is nilpotent and the reviewer doubts whether Corollary 3 is true.NEWLINENEWLINENEWLINEIn Lemma 7 it is proved that every \({\mathcal W}^*_k\)-group is restrained (a group \(G\) is said to be restrained if \(\langle x\rangle^{\langle y\rangle}\) is finitely generated for all \(x,y\in G\). If there is a bound on the number of generators of \(\langle x\rangle^{\langle y\rangle}\), then \(G\) is called strongly restrained.) The author in the proof of Theorem 4 based on the proof of Lemma 7 says that every \(\overline{\mathcal W}^*_k\)-group is strongly restrained, but the proof of Lemma 7 does not imply the latter claim and the reviewer doubts if it is true. Anyway, one can show, by another argument, that Theorem 4 remains true.NEWLINENEWLINENEWLINESome of the results of the paper have been proved with different methods by \textit{A. Abdollahi} and \textit{N. Trabelsi}, Quelques extensions d'un problème de Paul Erdős sur les groupes [to appear in Bull. Belg. Math. Soc. -- Simon Stevin].