Study on Cauchy Theorem as a Special Case of Sylow Theorem

Abstract

The Cauchy Group Theorem and the Sylow Theorem are foundations of the finite group theory. The Cauchy Group Theorem focuses on the existence of prime order elements in a finite group, whereas Sylow Theorems provide a more complete framework for the analysis of the structure and number of subgroups. A formal proof of the two theorems is given in this paper. The Cauchy Group Theorem is proved by proving the existence of an order  element in a finite group , here  is a prime divisor of , and  is the order of group . The Sylow Theorem is proved by existence, conjugacy and amount of Sylow -subgroups. In this process, the relation between two theorems is revealed. In other words, the Cauchy Group Theorem is a special case of Sylow Theorem, and the Cauchy Group Theorem is equivalent to the case  in which the order of group is . The Cauchy Group Theorem is used to classify finite simple groups and the Structure Theorem of finite abelian groups. The application of the Sylow Theorem is to classify and analyze the group structure.