Introduction of Group Action and Examples of the Action

Yuanzhe Liu

doi:10.54097/fpw6v553

Authors

Yuanzhe Liu

DOI:

https://doi.org/10.54097/fpw6v553

Keywords:

Group actions, Orbits and stabilizer; Group theory; Sylow’s theorem.

Abstract

The purpose of the essay is to provide a comprehensive understanding of key concepts in group theory, specifically focusing on group actions, orbits, and stabilizers. These important elements play a vital part in studying of algebraic structures and their symmetries. The author will begin by defining these concepts, explaining their significance, and illustrating them with practical examples. The discussion will include detailed definitions and explanations to build a solid foundation. In addition to these core concepts, the essay will introduce Sylow’s Theorems and Burnside’s Lemma. The author will explore how Sylow’s Theorems helps the number and structure of subgroups within a finite group, and how Burnside’s Lemma is used to count distinct objects under group actions. Through examples and detailed discussions, the author will demonstrate the practical applications of these theorems. Finally, readers are going to have a clear comprehension of how these mathematical tools are applied to solve problems and analyze group structures effectively. This works underscores the importance of group action in modern science.

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