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A NILPOTENCY CRITERION FOR SOME VERBAL SUBGROUPS

  • CARMINE MONETTA (a1) and ANTONIO TORTORA (a2) (a3)

Abstract

The word $w=[x_{i_{1}},x_{i_{2}},\ldots ,x_{i_{k}}]$ is a simple commutator word if $k\geq 2,i_{1}\neq i_{2}$ and $i_{j}\in \{1,\ldots ,m\}$ for some $m>1$ . For a finite group $G$ , we prove that if $i_{1}\neq i_{j}$ for every $j\neq 1$ , then the verbal subgroup corresponding to $w$ is nilpotent if and only if $|ab|=|a||b|$ for any $w$ -values $a,b\in G$ 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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The authors are members of National Group for Algebraic and Geometric Structures and their Applications (GNSAGA–INdAM).

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References

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