The order of the product of two elements in the periodic groups

TL;DR

This paper investigates the properties of element orders in periodic groups, establishing conditions under which certain subgroups are locally nilpotent and characterizing finite abelian groups through degree sums related to conjugacy classes.

Contribution

It introduces the set $LCM(G)$ in periodic groups and proves the subgroup generated by it is locally nilpotent in locally finite groups, and characterizes finite abelian groups via degree sums.

Findings

The subgroup generated by $LCM(G)$ is locally nilpotent in locally finite groups.

In finite groups, the degree sum condition characterizes abelian groups.

Provides a new perspective on element order relations in periodic groups.

Abstract

Let $G$ be a periodic group, and let $LCM(G)$ be the set of all $x\in G$ such that $o(x^nz)$ divides the least common multiple of $o(x^n)$ and $o(z)$ for all $z$ in $G$ and all integers $n$. In this paper, we prove that the subgroup generated by $LCM(G)$ is a locally nilpotent characteristic subgroup of $G$ whenever $G$ is a locally finite group. For $x,y\in G$ the vertex $x$ is connected to vertex $y$ whenever $o(xy)$ divides the least common multiple of $o(x)$ and $o(y)$. Let $Deg(G)$ be the sum of all $deg(g)$ where $g$ runs over $G$. We prove that for any finite group $G$ with $h(G)$ conjugacy classes, $Deg(G)=|G|(h(G)+1)$ if and only if $G$ is an abelian group.