Center of the n\(^{th}\) Dihedral Group and Centralizers of the Elements of the n\(^{th}\) Dihedral Group, D\(_n\), where 3 ≤ n ≤ 20, n ∈ Z\(^+\)

Aims: To derive general formulas for determining the center and centralizers of the elements of the nth dihedral group, Dn, where 3 ≤ n ≤ 20, and to evaluate their consistency and potential for generalization. Study Design: This is a computational and theoretical study in group theory employing definition-based and pattern-recognition methodologies. Place and Duration of Study: Department of Mathematics, College of Science, University of Eastern Philippines, Catarman, Northern Samar, Philippines, conducted between August 2023 and March 2024. Methodology: The study applied foundational definitions from abstract algebra to compute the center and centralizers of dihedral group of degree n, ​ for values of n is greater than or equal to three and less than or equal to twenty. A systematic pattern recognition method was used to derive general formulas distinguishing between even and odd values of n. Verification was conducted through direct computation and formula-based prediction to validate the observed patterns. Results: For odd values of n, the center Z(Dn​) consisted solely of the identity element, {e}, while for even n, it included both e and an/2. Centralizers of identity and specific elements like b, ai, and aib were derived and categorized, showing predictable, consistent patterns across tested values. For example, C(e) = DnC​ and C(ai) = {e, a, a2, ..., an−1} for rotational elements, while reflections exhibited smaller centralizers. Formulas held for all tested cases, with random tests beyond n = 20 also showing alignment, supporting the conjecture of general applicability for all n greater than or equal to three. Conclusion: The study offers efficient, formula-based computation of the center and centralizers of dihedral group of degree n​, significantly reducing time and effort compared to traditional methods. The derived expressions were validated for n is greater than or equal to three and less than or equal to twenty and are conjectured to be valid for all n greater than or equal to three, providing a useful tool for further research in group theory and its applications.