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.