## 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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