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A question of Paul Erdős and nilpotent-by-finite groups. (English) Zbl 0995.20020
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. J. C. Lennox, J. Wiegold, J. Aust. Math. Soc., Ser. A 31, 459-463 (1981; Zbl 0492.20019) and A. Abdollahi, B. Taeri, Commun. Algebra 27, No. 11, 5633-5638 (1999; Zbl 0942.20014)].
Here 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 \[ {\mathcal E}_k(n)\subseteq\overline\Omega_k(n)\subseteq\Omega_k(n)\subseteq\Omega_k(n+1)\subseteq\Omega_k(\infty) \] and \[ \overline{\mathcal U}_k(n)\subseteq{\mathcal U}_k(n)\subseteq\Omega_k(n)\text{ and }{\mathcal E}_k^*\subseteq\overline{\mathcal W}^*_k\subseteq{\mathcal W}^*_k\subseteq\Omega_k(\infty). \] 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 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 P. H. Kropholler [Proc. Lond. Math. Soc., III. Ser. 49, 155-169 (1984; Zbl 0537.20013)]. P. Longobardi and 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 J. S. Wilson [Bull. Lond. Math. Soc. 23, No. 3, 239-248 (1991; Zbl 0746.20018)]. O. Puglisi and 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.
Reviewer’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 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.
In 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.
Some of the results of the paper have been proved with different methods by A. Abdollahi and N. Trabelsi, Quelques extensions d’un problème de Paul Erdős sur les groupes [to appear in Bull. Belg. Math. Soc. – Simon Stevin].

MSC:
20F19 Generalizations of solvable and nilpotent groups
20F16 Solvable groups, supersolvable groups
20F18 Nilpotent groups
20E34 General structure theorems for groups
20E25 Local properties of groups
20F12 Commutator calculus
20F45 Engel conditions
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