# A criterion for metanilpotency of a finite group

## TL;DR

This paper establishes a criterion for metanilpotency in finite groups by linking the nilpotency of the lower central series' terms to a specific order condition on certain commutators.

## Contribution

It provides a new characterization of metanilpotency in finite groups based on the order behavior of $ ext{γ}_k$-commutators.

## Key findings

- The $k$th term of the lower central series is nilpotent if and only if $|ab|=|a||b|$ for coprime $ ext{γ}_k$-commutators.
- The criterion offers a practical test for metanilpotency in finite groups.
- The result deepens understanding of the structure of finite groups through commutator properties.

## Abstract

We prove that the $k$th term of the lower central series of a finite group $G$ is nilpotent if and only if $|ab|=|a||b|$ for any $\gamma_k$-commutators $a,b\in G$ of coprime orders.