A Note on Finite Nilpotent Groups

TL;DR

This paper characterizes finite nilpotent groups by a property relating element powers to subgroups of finite index, showing that such groups are precisely those where every element's power related to subgroup index lies within the subgroup.

Contribution

It establishes a new characterization of finite nilpotent groups based on the behavior of element powers relative to all subgroups of finite index.

Findings

Finite groups with the property are exactly the nilpotent groups.

The property holds iff the group is nilpotent.

Provides a new criterion for nilpotency in finite groups.

Abstract

It is well known that if $G$ is a group and $H$ is a normal subgroup of $G$ of finite index $k$, then $x^k \in H$ for every $x \in G$. We examine finite groups $G$ with the property that $x^k \in H$ for every subgroup $H$ of $G$, where $k$ is the index of $H$ in $G$. We prove that a finite group $G$ satisfies this property if and only if $G$ is nilpotent.