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Paper   IPM / M / 2327
School of Mathematics
  Title:   On a combinatorial problem in group theory
  Author(s):  B. Taeri
  Status:   Published
  Journal: Southeast Asian Bull. Math.
  Vol.:  26
  Year:  2003
  Pages:   1029-1039
  Supported by:  IPM
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 X0X , with 2 ≤ | X0 | ≤ n + 1 and a function f : {0, 1, 2,..., k} → X0 , with f (0) ≠ f (1) and non-zero integers t0 , t1 , ... , tk such that [xt00 ,xt11 , ... , xtkk ] = 1, where xi : = f (i), i = 0 ,..., k, and xjH whenever xtjjH, for some subgroup H ≠ 〈xtjj 〉 of G. If the integer k is fixed for every subset X we obtain the class Wk (n). Here we prove that

    1) Let GW (n), n a positive integer, be a finite group, p > n a prime divisor of the order of G, P a Sylow p-subgroup 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 finite-by-nilpotent.
    3) Let GWk (∞) be a finitely generated soluble group, then G is finite-by-(nilpotent of k-bounded class).

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