The order of the product of two elements in the 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.