Group Action, Orbit-Stabilizer Theorem and Burnside's Lemma

This paper offers a thorough exploration of group action, equivalence class, and their utilization in combinatorial counting problems. Starting with the fundamentals of group theory and set relations, this section encompasses definitions and proofs of elementary results. Then, it proceeds to provide a proof of Lagrange's Theorem by illustrating that orbits divide the set into distinct equivalence classes through the use of an equivalence relation within the set. Following Lagrange's Theorem, this paper establishes the Orbit-Stabilizer Theorem by demonstrating that stabilizers partition the entire group and proving that each stabilizer contains the same number of elements. This paper subsequently demonstrates Burnside's Lemma, a tool employed for determining the quantity of orbits under a particular group action. This is achieved by utilizing the Orbit-Stabilizer Theorem and more profound insights derived from an equivalence relation concerning partitions. Therefore, this approach can be utilized to tackle the face coloring issue on a cube. Analyzing the various operations and determining the number of configurations they resolve. In conclusion, this study places these findings in a broader algebraic framework, emphasizing their ties to Pólya's Enumeration Theorem and computational group algorithms.