Gröbner-Shirshov bases for the Coxeter groups of the types $G_2,F_4,E_6,E_7$ and $E_8$

Abstract: The author is mainly interest in the Gr\"{o}bner-Shirshov bases of finite Coxeter groups. It is known that the finite Coxeter groups are classified in terms of Coxeter-Dynkin diagrams. Under the fixed order, it is worth mention that the presentation of group determines the Gr\"{o}bner-Shirshov bases of group. In this paper, the author rearranges the generators, marks them on Coxeter-Dynkin diagrams, and gets a simple presentation of the Gr\"{o}bner-Shirshov bases for Coxeter groups of types $G_2, F_4, E_6$ and $E_7$. This article also gives the Gr\"{o}bner-Shirshov basis of Coxeter group of type $E_8$.