THE ORDER GRAPHS OF GROUPS

Let $G$ be a group. The order graph of $G$ is the (undirected)graph $Gamma(G)$,those whose vertices are non-trivial subgroups of $G$ and two distinctvertices $H$ and $K$ are adjacent if and only if either$o(H)|o(K)$ or $o(K)|o(H)$. In this paper, we investigate theinterplay between the group-theoretic properties of $G$ and thegraph-theoretic properties of $Gamma(G)$. For a finite group$G$, we show that $Gamma(G)$ is a connected graph with diameter at mosttwo, and     $Gamma(G)$ is a complete graph ifand only if $G$ is a $p$-group for some prime number $p$. Furthermore,it is shown that $Gamma(G)=K_5$ if and only if either$Gcong C_{p^5}, C_3times C_3$, $C_2timesC_4$ or $Gcong Q_8$.