A nilpotency criterion for some verbal subgroups

2018/12/05 by Monetta, Carmine, Tortora, Antonio
#20F18 #20F45 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1812.02123

Abstract

The word w=[x_i1,x_i2,…,x_ik] is a simple commutator word if k≥ 2, i₁≠ i₂ and i_j∈ \1,…,m\, for some m>1. For a finite group G, we prove that if i₁ ≠ 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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