THE ORDER GRAPHS OF GROUPS

Let GG be a group. The order graph of GG is the (undirected) graph Γ(G)Γ(G), those whose vertices are non-trivial subgroups of GG and two distinct vertices HH and KK are adjacent if and only if either o(H)|o(K)o(H)|o(K) or o(K)|o(H)o(K)|o(H). In this paper, we investigate the interplay between the group-theoretic properties of GG and the graph-theoretic properties of Γ(G)Γ(G). For a finite group GG, we show that Γ(G)Γ(G) is a connected graph with diameter at most two, and Γ(G)Γ(G) is a complete graph if and only if GG is a pp-group for some prime number pp. Furthermore, it is shown that Γ(G)=K5Γ(G)=K5 if and only if either G≅Cp5,C3×C3G≅Cp5,C3×C3, C2×C4C2×C4 or G≅Q8G≅Q8.