A Sufficient Condition for Nilpotency of the Nilpotent Residual of a Finite Group

TL;DR

This paper establishes a sufficient condition involving the orders of certain commutators in a finite group that guarantees the nilpotency of its nilpotent residual, contributing to the understanding of group structure.

Contribution

It introduces a new condition on powers of $ ext{delta}_1^*$-commutators that ensures the nilpotency of the group's nilpotent residual.

Findings

The nilpotent residual $ ext{γ}_ ext{∞}(G)$ is nilpotent under the given condition.

The condition relates the orders of powers of specific commutators with coprime orders.

Provides a structural criterion for nilpotency in finite groups.

Abstract

Let $G$ be a finite group with the property that if $a,b$ are powers of $\delta_1^*$-commutators such that $(|a|,|b|)=1$, then $|ab|=|a||b|$. We show that $\gamma_{\infty}(G)$ is nilpotent.