Theory and Applications of Orbits and Stabilizer in Group Action

Abstract

The algebraic system of groups has become the focus of modern algebra courses because it is an important tool for studying symmetry problems. However, it is difficult for students to understand groups because it is difficult for them to understand how it can be used as a tool to study symmetry problems. In this article, the author presents a series of proofs, importance, and examples of the definition of orbits and stabilizer, the proof of useful claim, and the definition of orbits and stabilizer as an important part of swarm action. The characteristics of its derivative theorem orbits-stabilizers theorem are analyzed. Besides, the article also includes the application of Orbits-Stabilizer theorem to the Cauchy's Theorem, Lagrange's Theorem and Burnside's Lemma theorem by giving several definition, examples, and proof. Finally, this paper not only reviews the previous understanding and summary of the role of groups, but also contains the author's prospects for orbits and stabilizers in other fields.