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 eithero(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 groupG, 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_۵ if and only if eitherG\cong C_{p^۵}, C_۳\times C_۳, C_۲\timesC_۴ or G\cong Q_۸.