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| Paper IPM / M / 2327 |
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| Abstract: | |
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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 X0 ⊆ X , 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 xj ∈ H whenever xtjj ∈ H, for some
subgroup H ≠ 〈xtjj 〉 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 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 G ∈ W∗k (∞) be a finitely generated soluble group, then G is finite-by-(nilpotent of k-bounded class). Download TeX format |
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