The word w=[x_{i_1},x_{i_2}, …, x_{i_k}] is a simple commutator word if k ≥ 2, i_1 ≠ i_2 and i_j ∈ {1, …, m}, for some m > 1. For a finite group G, we prove that if i_1≠i_j for every j≠1, then the verbal subgroup corresponding to w is nilpotent if and only if |ab|=|a||b| for any w-values a,b∈G of coprime orders. We also extend the result to a residually finite group G, provided that the set of all w-values in G is finite.

A nilpotency criterion for some verbal subgroups

Antonio Tortora
2019

Abstract

The word w=[x_{i_1},x_{i_2}, …, x_{i_k}] is a simple commutator word if k ≥ 2, i_1 ≠ i_2 and i_j ∈ {1, …, m}, for some m > 1. For a finite group G, we prove that if i_1≠i_j for every j≠1, then the verbal subgroup corresponding to w is nilpotent if and only if |ab|=|a||b| for any w-values a,b∈G of coprime orders. We also extend the result to a residually finite group G, provided that the set of all w-values in G is finite.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11591/408587
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