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 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.