THE ORDER GRAPHS OF GROUPS

Abstract

Let $G$ be a group. The order graph of $G$ is the (undirected)<br />graph $Gamma(G)$,<br />those whose vertices are non-trivial subgroups of $G$ and two distinct<br />vertices $H$ and $K$ are adjacent if and only if either<br />$o(H)|o(K)$ or $o(K)|o(H)$. In this paper, we investigate the<br />interplay between the group-theoretic properties of $G$ and the<br />graph-theoretic properties of $Gamma(G)$. For a finite group<br />$G$, we show that $Gamma(G)$ is a connected graph with diameter at most<br />two, and $Gamma(G)$ is a complete graph if<br />and only if $G$ is a $p$-group for some prime number $p$. Furthermore,<br />it is shown that $Gamma(G)=K_5$ if and only if either<br />$Gcong C_{p^5}, C_3times C_3$, $C_2times<br />C_4$ or $Gcong Q_8$.