Advanced Insights into Finite Group Structures and Their Symmetries
DOI:
https://doi.org/10.53573/rhimrj.2025.v12n7.019Keywords:
Finite Group Structures, Lagrange’s theorem, Cauchy’s theorem, geometric phenomenaAbstract
Finite group theory has long been a cornerstone of abstract algebra, offering a structured framework to investigate symmetry, order, and transformation across diverse mathematical and scientific domains. This paper provides an in-depth exploration of finite group structures, focusing on the intricate relationships among subgroups, quotient groups, and their associated symmetries. Building upon foundational results such as Lagrange’s theorem, Cauchy’s theorem, and Sylow’s theorems, we present advanced perspectives on the classification and behavior of finite groups. Special attention is given to group actions, orbit-stabilizer relations, conjugacy classes, and character theory, as they illuminate the hidden symmetries that govern both algebraic and geometric phenomena. In addition to theoretical analysis, the paper examines the role of finite groups in modeling symmetry across multiple disciplines. Applications in crystallography, molecular structures, and quantum mechanics demonstrate the relevance of group symmetries in the physical sciences, while contributions to cryptography, coding theory, and combinatorial optimization highlight their computational and applied significance. The integration of computational group theory and algorithmic methods has further expanded the frontiers of research, enabling the analysis of large and complex groups with efficiency and precision. Ultimately, this study underscores the unifying role of finite group theory in bridging pure mathematics with interdisciplinary applications. By revisiting classical theorems through modern frameworks and computational insights, we seek to advance a deeper understanding of finite group structures and their symmetries, while also charting pathways for future research in both theoretical and applied contexts.
References
Rotman, J. J. (2012). An Introduction to the Theory of Groups (4th ed.). Springer.
Huppert, B. (1967). Endliche Gruppen I. Springer-Verlag.
Serre, J.-P. (1977). Linear Representations of Finite Groups. Springer.
Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A., & Wilson, R. A. (1985). Atlas of Finite Groups. Oxford University Press.
Isaacs, I. M. (2008). Finite Group Theory. American Mathematical Society.
Dixon, J. D., & Mortimer, B. (1996). Permutation Groups. Springer-Verlag.
Cameron, P. J. (1999). Permutation Groups and Applications. Cambridge University Press.
Wielandt, H. (1964). Finite Permutation Groups. Academic Press.
Aschbacher, M. (2000). Finite Group Theory (2nd ed.). Cambridge University Press.
Gorenstein, D. (1980). Finite Groups. Chelsea Publishing.
Carter, R. W. (1993). Finite Groups of Lie Type: Conjugacy Classes and Complex Characters. John Wiley & Sons.
Babai, L., & Pyber, L. (1995). Permutation Groups and Combinatorial Structures. Journal of Combinatorial Theory, Series A, 69(2), 173–188.
Lubotzky, A., & Segal, D. (2003). Subgroup Growth. Birkhäuser.
Seress, Á. (2003). Permutation Group Algorithms. Cambridge University Press.
Cameron, P. J., & Cohen, A. M. (1992). On the Number of Conjugacy Classes in a Finite Group. Journal of Algebra, 154(2), 343–355.
Dixon, J. D. (1996). The Probability of Generating the Symmetric Group. Mathematics of Computation, 65(213), 1235–1240.
