A Sufficient Condition for Nilpotency of the Commutator Subgroup

TL;DR

This paper proves that for a finite group where the order of the product of certain commutators equals the product of their orders, the commutator subgroup is necessarily nilpotent.

Contribution

It establishes a sufficient condition involving coprime order commutators for the nilpotency of the commutator subgroup in finite groups.

Findings

The commutator subgroup is nilpotent under the given condition.

The condition relates the order of products of coprime order commutators to their individual orders.

Provides a new criterion for nilpotency in finite groups.

Abstract

Let $G$ be a finite group with the property that if $a,b$ are commutators of coprime orders, then $|ab|=|a||b|$. We show that $G'$ is nilpotent.