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Paper IPM / M / 2327  


Abstract:  
Let n be a positive integer or infinity (denote ∞ ). We
denote by W^{∗} (n) the class of groups G such that, for
every subset X of G of cardinality n + 1, there exist a
positive integer k, and a subset X_{0} ⊆ X , with 2 ≤  X_{0}  ≤ n + 1 and a function f : {0, 1, 2,..., k} → X_{0} , with f (0) ≠ f (1) and nonzero
integers t_{0} , t_{1} , ... , t_{k} such that [x^{t0}_{0} ,x^{t1}_{1} , ... , x^{tk}_{k} ] = 1, where x_{i} : = f (i), i = 0 ,..., k, and x_{j} ∈ H whenever x^{tj}_{j} ∈ H, for some
subgroup H ≠ 〈x^{tj}_{j} 〉 of G. If the
integer k is fixed for every subset X we obtain the class
W^{∗}_{k} (n). Here we prove that
1) Let G ∈ W^{∗} (n), n a positive integer, be a finite group, p > n a prime divisor of the order of G, P a Sylow psubgroup of G. Then there exists a normal subgroup K of G such that G = P ×K. 2) A finitely generated soluble group has the property W^{∗} ( ∞) if and only if it is finitebynilpotent. 3) Let G ∈ W^{∗}_{k} (∞) be a finitely generated soluble group, then G is finiteby(nilpotent of kbounded class). Download TeX format 

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