Conciseness of coprime commutators in finite groups

TL;DR

This paper proves that in finite groups, the subgroup generated by coprime $ ext{commutators}$ has bounded order based on the size of the set of such commutators, extending classical results on word conciseness.

Contribution

It establishes bounds on the order of subgroups generated by coprime commutators, generalizing the classical conciseness theorem for specific words in finite groups.

Findings

Order of subgroup bounded by set size of coprime commutators

Extends classical conciseness results to coprime commutators

Provides new bounds in finite group theory

Abstract

Let $G$ be a finite group. We show that the order of the subgroup generated by coprime $\gamma_k$-commutators (respectively $\delta_k$-commutators) is bounded in terms of the size of the set of coprime $\gamma_k$-commutators (respectively $\delta_k$-commutators). This is in parallel with the classical theorem due to Turner-Smith that the words $\gamma_k$ and $\delta_k$ are concise.