# A note on the order graph of a group

Document Type : Research Paper

Author

University of Jiroft

Abstract

The order graph of a group $G$, denoted by $\Gamma^*(G)$, is a graph whose vertices are subgroups of $G$ and two distinct vertices $H$ and $K$ are adjacent if and only if $|H|\big{|}|K|$ or $|K|\big{|}|H|$.
In this paper, we study the connectivity and diameter of this  graph. Also we give a relation between the order graph and prime  graph of a group.

Keywords

#### References

[1] J. A. Bondi, J. S. Murty, Graph theory with applications, American Elsevier Publishing Co, INC, 1997.
[2] Y. Chen, On Thompson’s conjecture, J. Algebra 15 (1996), 184-193.
[3] J. A. Gallian, Contemporary Abstract Algebra, D. C. Heath and company, 1994.
[4] B. Huppert, Character Theory of Finite Groups, De Gruyter Expositions in Mathematics, New York 1998.
[5] Sh. Payrovi, H. Pasebani, The Order Graphs of Groups, J Algebraic Structures and Their Applications, 1
(no 1) ( 2014 ), 1-10.
[6] J. S. Wiliams, Prime Graph Components of Finite Groups, J. Algebra 69 (1981), 487-513.