Cauchy Theorem, Sylow Theorems, and Orbit-Stabilizer Theorem in Group Theory

Abstract

With the rapid development of modern mathematics, math researchers have an increasing demand to take the advantage of group theory in the latest field. The group action is one of the essential parts of the group theory. In order to have a better understand of group action, this paper will describe the insight development and logic in it step by step. For this purpose, three essential theorems, which are Cauchy theorem, Sylow theorems, and orbit-stabilizer theorem, are chosen as representative examples. The proof and applications in some fields of modern science of these three theorems are discussed. In the proof, this paper emphasizes that group action can be used conveniently to solve the problem in group theory. In the application, this paper includes as many fields as possible to attach importance to group action. This paper is expected to give the opinion of how the field of group action is established and its far-reaching influence to the modern science.