International Journal For Multidisciplinary Research

E-ISSN: 2582-2160   •   Impact Factor: 9.24

A Widely Indexed Open Access Peer Reviewed Multidisciplinary Bi-monthly Scholarly International Journal

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A new perspective on Cyclic Group Structure and its Implications

Author(s) Mr. PAVAN KUMAR H
Country India
Abstract Cyclic groups form a foundational concept in abstract algebra, serving as essential building blocks for understanding broader group structures and algebraic systems. This paper presents a new perspective on the structure of cyclic groups by exploring their intrinsic properties through an algebraic and geometric lens. The study reinterprets the generation process, subgroup hierarchy, and element order distribution within cyclic groups, revealing novel connections between arithmetic progressions and group homomorphisms. Furthermore, it examines the implications of these structural insights for applications in number theory, coding theory, and cryptography, particularly in modular arithmetic and discrete logarithmic problems. The proposed framework not only enhances the conceptual understanding of cyclic group dynamics but also provides an alternative approach to classifying finite and infinite cyclic groups. By integrating classical theorems with new analytical tools, this work offers a unifying perspective that bridges traditional group theory with emerging computational and theoretical advancements, paving the way for future research on group symmetry and algebraic structure optimization.
Keywords Cyclic groups; Group theory; Algebraic structure; Generators and subgroups; Finite and infinite groups; Group homomorphism; Modular arithmetic; Number theory; Symmetry; Cryptography applications; Abstract algebra; Structural analysis; Group classification; Mathematical optimization; Discrete logarithm.
Field Mathematics
Published In Volume 7, Issue 6, November-December 2025
Published On 2025-12-24
DOI https://doi.org/10.36948/ijfmr.2025.v07i06.64395
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E-ISSN 2582-2160
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