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| Paper IPM / M / 624 |
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| Abstract: | |
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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,y ∈ X and integers t0, t1,…, tk such that [x0t0, x1t1, …,xktk]=1, where xi ∈ {x,y}, i=0,1,…, k,x0 ≠ x1. If the integers t0, t1, …, tk are the same for any
subset X of G, we say that G is in the class
―Ωk(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 x ∈ X, y ∈ Y such that [x0, x1, x2t2..., xktk]=1,
where xi ∈ {x,y}, x0 ≠ x1, i=2,3,…, k. Here we
prove that
1 If G ∈ Uk(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 G ∈ ―Ωk(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. Download TeX format |
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