The Nilpotency Criterion for the Derived Subgroup of a Finite Group

TL;DR

This paper establishes a criterion for the nilpotency of the derived subgroup in finite groups based on a specific inequality involving primary commutators of coprime orders.

Contribution

It provides a new necessary and sufficient condition for the derived subgroup of a finite group to be nilpotent, linking group structure to commutator order relations.

Findings

Derived subgroup is nilpotent iff the inequality holds for all primary commutators of coprime orders.

Characterizes nilpotency of the derived subgroup through order relations of primary commutators.

Offers a criterion that can be checked to determine nilpotency in finite groups.

Abstract

It is proved that the derived subgroup of a finite group is nilpotent if and only if $|ab|\ge |a||b|$ for all primary commutators $a$ and $b$ of coprime orders.