The Finite Matrix Group Over Field

Assume that R is commutative ring which has identity element. GL(n,R) isa set of invertible matrix over R which order is n. Then GL(n,R) act as a groupunder multiplication. Generally, we call GL(n,R) general linear group over R whichorder is n.General linear group is very important group and play important roles in theentire process of group theory research. For example,classical groups, the theory ofgroup representations, the theory of abstract group, crystallography, etc. The rela-tionship between general linear group and other type of group make many scholarsresearch its properties. This paper deals with the study of the structure of generallinear group GL(n,R)'s the finite subgroup over Q.In Chapter 1, we mainly introduce the works related to this thesis and contentand method to study in this thesis.In Chapter 2, we give some necessary preliminaries, including some basic con-ceptions , lemmas.In Chapter 3, we deal with the order's upper bound of a finite subgroup ofGL(n,Q). Through searching the structure of elementary abelian 2-subgroup ofGL(n,Z), we figure out a order's upper bound of a finite subgroup of GL(n,Q),improve the result in the literature[35].In Chapter 4, we focus on the structure of finite subgroup of GL(n,Q), whenn is relatively small. By using the knowledge of structure of finite subgroup and ele-mentary theory of numbers, we figure out the finite subgroup's structure of GL(2,Z)and GL(2,Q). Furthermore, we deal with the periodic element's conjugate class ofGL(n,Q), when n = 3,4.