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This is a Computational Study study (also classified as: Theoretical Study).

September 10, 2025Open Access

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\ (^+\)

JGJessie T. Gento RARogie Mhej R. Añonuevo SMShiela F. Mendez

Key Points

The center of dihedral groups for odd values of n consists of the identity element only.
For even values of n, the center includes both the identity element and another distinct element.
The findings show consistent patterns in centralizers across specific elements in the dihedral group.
Validation through tests for n values beyond 20 supports conjectured formulas' general applicability.

Abstract

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.

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