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.File in questo prodotto:
Non ci sono file associati a questo prodotto.
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
