The Investigation of Orbit-Stabilizer Theorem and Its Applications

Group theory is an important area of modern mathematics, and the orbit-stabilizer theorem is a significant conclution in all of the group theory as it is the basic of many other conclusions. This article introduces theory of group action and the definition of orbit and stabilizer as a background for orbit stabilizer theorem. The theorem itself was then stated with proof. Orbit-stabilizer theorem says that when a finite group is acting on a finite set, the orbit of any element of the set must have the same size with number of left cosets of stabilizer of the element in the group. The proof was then given using the fact that two sets with a bijection between them must have same size. Examples of applications in the proof of Cauchys theorem and Burnsides lemma were given to show the importance of orbit stabilizer theorem in Group theory. Orbit-stabilizer theorem is a central part in the proof of both of those results.