“School of Mathematics”
Back to Papers HomeBack to Papers of School of Mathematics
Paper IPM / M / 624  


Abstract:  
Let n be a positive integer or infinity (denoted ∞), k
a positive integer. We denote by Ω_{k}(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 t_{0}, t_{1},…, t_{k} such that [x_{0}^{t0}, x_{1}^{t1}, …,x_{k}^{tk}]=1, where x_{i} ∈ {x,y}, i=0,1,…, k,x_{0} ≠ x_{1}. If the integers t_{0}, t_{1}, …, t_{k} are the same for any
subset X of G, we say that G is in the class
―Ω_{k}(n). The class U_{k}(n) is defined
exactly as Ω_{k}(n) with the additional conditions
x_{i}^{ti} ≠ 1. Let t_{2},t_{3},…, t_{k} 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 [x_{0}, x_{1}, x_{2}^{t2}..., x_{k}^{tk}]=1,
where x_{i} ∈ {x,y}, x_{0} ≠ x_{1}, i=2,3,…, k. Here we
prove that
1 If G ∈ U_{k}(2) is a finitely generated soluble group. then G is nilpotent. 2 If G ∈ Ω_{k}(∞) is a finitely generated soluble group. then G is nilpotentbyfinite. 3 If G ∈ ―Ω_{k}(n), n a positive integer, is a finitely generated residually finite group. then G is nilpotentbyfinite. 4 If G is an infinite ―W_{k}^{*}group in which every nontrivial finitely generated subgroup has a nontrivial finite quotient. then G is nilpotentbyfinite. Download TeX format 

back to top 