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Paper   IPM / M / 624
School of Mathematics
  Title:   A question of Paul Erdos and nilpotent-by-finite groups
  Author(s):  B. Taeri
  Status:   Published
  Journal: Bull. Aust. Math. Soc.
  No.:  2
  Vol.:  64
  Year:  2001
  Pages:   245-254
  Supported by:  IPM
Let n be a positive integer or infinity (denoted ∞), k a positive integer. We denote by Ωk(n) the class of groups G such that, for every subset X of G of cardinality n+1, there exist distinct elements x,yX and integers t0, t1,…, tk such that [x0t0, x1t1, …,xktk]=1, where xi ∈ {x,y}, i=0,1,…, k,x0x1. If the integers t0, t1, …, tk are the same for any subset X of G, we say that G is in the class k(n). The class Uk(n) is defined exactly as Ωk(n) with the additional conditions xiti ≠ 1. Let t2,t3,…, tk be fixed integers. We denote by W*k the class of all groups G such that for any infinite subsets X and Y there exist xX, yY such that [x0, x1, x2t2..., xktk]=1, where xi ∈ {x,y}, x0x1, i=2,3,…, k. Here we prove that

    1 If GUk(2) is a finitely generated soluble group. then G is nilpotent.
    2 If G ∈ Ωk(∞) is a finitely generated soluble group. then G is nilpotent-by-finite.
    3 If Gk(n), n a positive integer, is a finitely generated residually finite group. then G is nilpotent-by-finite.
    4 If G is an infinite Wk*-group in which every non-trivial finitely generated subgroup has a non-trivial finite quotient. then G is nilpotent-by-finite.

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