Bounding the Exponent of a Verbal Subgroup

TL;DR

This paper investigates bounds on the exponent of verbal subgroups in finite groups under certain conditions, proving the conjecture for specific words and establishing bounds based on subgroup properties.

Contribution

It proves the conjecture for the nth Engel word and a specific commutator word, and generalizes bounds for verbal subgroups based on subgroup exponents.

Findings

Bounded the exponent of w(G) for Engel and specific commutator words.

Established a general bound for verbal subgroups based on subgroup exponents.

Provided explicit conditions under which the conjecture holds.

Abstract

We deal with the following conjecture. If w is a group word and G is a finite group in which any nilpotent subgroup generated by w-values has exponent dividing e, then the exponent of the verbal subgroup w(G) is bounded in terms of e and w only. We show that this is true in the case where w is either the nth Engel word or the word [x^n,y_1,y_2,...,y_k] (Theorem A). Further, we show that for any positive integer e there exists a number k=k(e) such that if w is a word and G is a finite group in which any nilpotent subgroup generated by products of k values of the word w has exponent dividing e, then the exponent of the verbal subgroup w(G) is bounded in terms of e and w only (Theorem B).