The word w = [xi1 , xi2 , : : : ; xik] is a simple commutator word if k ≤ 2; i1 ≠ i2 and ij i {1,...m} for some m>1. For a finite group G, we prove that if i1 ≠ ij for every j ≠ 1, then the verbal subgroup corresponding to is nilpotent if and only if |ab|=|a||b| for any w-values a,b iG 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
Monetta C.;
2019
Abstract
The word w = [xi1 , xi2 , : : : ; xik] is a simple commutator word if k ≤ 2; i1 ≠ i2 and ij i {1,...m} for some m>1. For a finite group G, we prove that if i1 ≠ ij for every j ≠ 1, then the verbal subgroup corresponding to is nilpotent if and only if |ab|=|a||b| for any w-values a,b iG 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:
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