On the Proof of Sylow’s Theorems in Group Theory

Abstract

The Sylow’s theorems are significant principles for analysis of special subgroups of a finite group, and they are significant theories in finite group research area. It is firmly established that every group owns more than one Sylow group pertaining to every prime factor of . The purpose of this article is to summarize and generalize the existing research progress of Sylow’s theorems. A summary of them will help more people comprehend and apply group theory to analyze problems. In this paper, the basic information of three distinct Sylow’s theorems are introduced, including their definitions and detailed proof procedures. After that, some examples and application that are related to the Sylow’s theorems are shown one by one. After that, the relationship between the Sylow’s theorems to that of the orbit-stabilizer theorem is discussed. This work will potentially stimulate more research efforts on the basis theorems in group theory.