چکیده انگلیسی:
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 distinct vertices $H$ and $K$ are adjacent if and only if either $o(H)|o(K)$ or $o(K)|o(H)$. In this paper, we investigate the interplay between the group-theoretic properties of $G$ and the graph-theoretic properties of $Gamma(G)$. For a finite group $G$, we show that $Gamma(G)$ is a connected graph with diameter at most two, and $Gamma(G)$ is a complete graph if and 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 $Gcong C_{p^5}, C_3times C_3$, $C_2times C_4$ or $Gcong Q_8$.
خبرنامه
برای ثبت نام در خبرنامه و دریافت خبرنامه ایمیل خود را وارد نمایید.