  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.
