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On a class of residually finite groups. (English) Zbl 1061.20035

Let \(n,k\) be positive integers and \(t_0,t_1,\dots,t_k\) be non-zero integers. In the paper under review the author considers the following classes of groups which are defined by some combinatorial Engel like conditions on certain subsets of a group.
The class \(\overline W_k(n)\) is defined as the class of groups \(G\) in which, for every subset \(X\) of \(G\) of cardinality \(n+1\), there exist a subset \(X_0\subseteq X\), with \(2\leq|X_0|\leq n+1\), and a function \(f\colon\{0,1,2,\dots,k\}\to X_0\), with \(f(0)\not=f(1)\) such that \([x_0^{t_0},x_1^{t_1},\dots,x_k^{t_k}]=1\) where \(x_i:=f(i)\), \(i=0,1,\dots,k\). – The class \(\overline W_k^*(n)\) is defined exactly as \(\overline W_k(n)\), with additional conditions “\(x_j\in H\) whenever \(x_j^{t_j}\in H\), where \(\langle x_j^{t_j}\rangle\not=H\leq G\)”.
For a finitely generated residually finite group \(G\), the author proves the following results:
Theorem A. If \(G\in \overline W_k(n)\), then \(G\) has a normal nilpotent subgroup \(N\) with finite index such that the nilpotency class of \(N/N_t\) is bounded by a function of \(k\), where \(N_t\) is the torsion subgroup of \(N\).
Theorem B. Let \(G\in\overline W_k^*(n)\) be a \(d\)-generated group. Then \(G\) has a normal nilpotent subgroup whose index and the nilpotency class is bounded by a function of \(k,n,t_0,t_1,\dots,t_k\).
The origin of considering such classes of groups is a problem posed by Paul Erdős which has been answered by B. H. Neumann [see J. Aust. Math. Soc., Ser. A 21, 467-472 (1976; Zbl 0333.05110)]. There are several papers concerning such classes of groups [see for example J. C. Lennox and J. Wiegold, J. Aust. Math. Soc., Ser. A 31, 459-463 (1981; Zbl 0492.20019), P. Longobardi, M. Maj and A. H. Rhemtulla, Commun. Algebra 20, No. 1, 127-139 (1992; Zbl 0751.20020), A. Abdollahi and N. Trabelsi, Bull. Belg. Math. Soc. - Simon Stevin 9, No. 2, 205-215 (2002; Zbl 1041.20022), and the reviewer, Houston J. Math. 27, No. 3, 511-522 (2001; Zbl 0999.20028)].

MSC:

20F45 Engel conditions
20E26 Residual properties and generalizations; residually finite groups
20E07 Subgroup theorems; subgroup growth
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