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Research Paper | Mathematics | India | Volume 12 Issue 8, August 2023
Exploring Properties and Isomorphisms of Automorphism Groups in Regular Covering Spaces
Abstract: The aim of this article is mainly focus on studying the properties of the automorphism group of a regular covering space. Regular covering spaces are those in which the isotropy subgroup corresponds to any point of the space is a normal subgroup of the fundamental group of the given topological space. By considering homomorphisms and autmorphisms of the covering spaces of a topological space, the details about all possible covering spaces of a given space can be obtained. The detailed study of First Poincare group is obtained from [1], [4] [6]& [10] . The theory of covering spaces and its properties were included in [3], [7]. For discussing about regular covering spaces, it is essential to define the fundamental group action on fibre set and we show that this action is transitive. Here we define an automorphism for a covering space and show that the set of all automorphisms forms a group under composition of transformations. Then discussion is about some properties of the automorphism group for a regular covering space. We also define an automorphism group for fibre set, by taking the restriction of the defined automorphism of regular covering space to fibre set. Also we show that the two automorphism groups so defined are isomorphic and we analyse the results.
Keywords: Regular Covering space, Fundamental group, Homogenius space, Transitive action, Homotopy, Auto morph ism, fibre set, group action
Edition: Volume 12 Issue 8, August 2023,
Pages: 2440 - 2444