Isolated subgroups of finite abelian groups

TL;DR

This paper characterizes isolated subgroups within finite abelian groups, revealing their structure through a novel connection to the sum of element orders, enhancing understanding of subgroup isolation.

Contribution

It provides a complete description of isolated subgroups in finite abelian groups using a new approach based on element order sums.

Findings

Characterization of isolated subgroups in finite abelian groups

Connection established between isolated subgroups and sum of element orders

Provides a method to identify isolated subgroups using element order functions

Abstract

We say that a subgroup $H$ is isolated in a group $G$ if for every $x\in G$ we have either $x\in H$ or $\langle x\rangle\cap H=1$. In this short note, we describe the set of isolated subgroups of a finite abelian group. The technique used is based on an interesting connection between isolated subgroups and the function sum of element orders of a finite group.