Topology and Automorphism Structures in General Linear Group.

Abstract

The general linear group, as a significant topic in algebra and topology, has a wide-ranging research background and application value. With the continuous advancement of mathematical research, the combination of topological structures and automorphism structures has become a key entry point for exploring the properties of the general linear group. This paper investigates the topological properties of general linear group GL(n,R) , specifically focusing on compactness, connectedness and the fundamental group, and the automorphism. The first part of the study is dedicated to a detailed analysis of these properties, providing new insights into the topological structural characteristics of GL(n,R) . This paper scrutinized the homotopy type within it, giving proof to the fundamental group of GL(n,R) , which is showed to be isomorphism to a trivial group. In the second half, This paper introduce and examine the function φ (G) = {A∈GL(n,R) |G• A = G} , which is used to identify the automorphism group of a dense subgroup G of Ｒⁿ . Specifically, the automorphism group of Ｑⁿ is investigated. This paper show that the automorphism group of it is isomorphic to the subgroup GL(n,Q) of general linear group GL(n,R) . Through this function, this paper offer an innovative approach to understanding the automorphisms within general linear groups, revealing the deep connections between algebraic and topological aspects of these groups.