On the Exponent of a Verbal Subgroup in a Finite Group

TL;DR

This paper proves that in finite groups, the exponent of a verbal subgroup generated by a multilinear commutator word is bounded by a function of e and w, given certain conditions on nilpotent subgroups.

Contribution

It establishes a bound on the exponent of verbal subgroups in finite groups based on properties of nilpotent subgroups generated by w-values.

Findings

Exponent of w(G) is bounded in terms of e and w.

Bound applies when nilpotent subgroups generated by w-values have exponent dividing e.

Provides a new understanding of the structure of verbal subgroups in finite groups.

Abstract

Let w be a multilinear commutator word. We prove that if e is a positive integer and G is a finite group in which any nilpotent subgroup generated by w-values has exponent dividing e then the exponent of the corresponding verbal subgroup w(G) is bounded in terms of e and w only.