Authors :
B. I. Ita; P. C. Ukachukwa; N. I. Ntoni; Charity Amanyi
Volume/Issue :
Volume 10 - 2025, Issue 12 - December
Google Scholar :
https://tinyurl.com/5bm24ucm
Scribd :
https://tinyurl.com/mw4acwx2
DOI :
https://doi.org/10.38124/ijisrt/25dec585
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Abstract :
In group theory, the cyclic group is a fundamental and seriously studied and understood classes, and is a corner
stone in the study of algebraic structures. This paper studies several generalized characteristics of cyclic group with the aim
of extending fundamental results and their implications within a broader context and applications to chemistry. We looked
at classical properties of cyclic groups, their generation, structure and subgroup behavior. We also explore generalizations
such as the decomposition of finite abelian into cyclic subgroup and their behavior under direct product constructions, and
the role of cyclicity in automorphism groups and homomorphic images. Using proof-based analysis, we show that these
generalized properties reveal deeper structural insights and enable a quick understanding of algebraic systems. Applications
to chemistry are also discussed to highlight the practical relevance of cyclic group theory. The paper concludes the directions
to future research on cyclic groups in some complex algebraic systems.
References :
- F. A. Cotton, and G. Wilkinson. (2021). Advanced Inorganic Chemistry, Wiley.
- D. M. Bishop. (2020). Group Theory and Chemistry. Dover publications.
- P. W. Atkins and R. Friedman. (2023). Molecular Quantum Mechanics. Oxford University Press.
- M. Hargittai and I. Hargittai. (2021). Symmetry through eyes of a Chemist. Springer.
- M. Ziolkowski. (2022). Applications of symmetry and group theory in Computational Chemistry. Journal of molecular modelling, 28 *5), 141-148.
- K. Balasubramanian. (2021). Group Theory and Molecular symmetry in Modern Quantum Chemistry. International Reviews in Physical Chemistry. 40 (1), 1-53.
- M. Tinkham. (2020). Group Theory and quantum Mechanics. Dover Publications.
- T. Solymosi and L. Nemes (2022). Group-Theoretical Analysis of Vibrational Spectra of Cyclic molecules. Spectrochimica Acta Part A: molecular and Biomolecular Spectroscopy. 273, 121026-121054.
- M. Senechal. (2021). Crystalline Symmetries: An Informal Mathematical Introduction. Dover.
- C. Liu. (2023). Cyclic Group Symmetry in Supramolecular Self-Assembly. Chemical Society Reviews. 52(12), 4931-4954.
In group theory, the cyclic group is a fundamental and seriously studied and understood classes, and is a corner
stone in the study of algebraic structures. This paper studies several generalized characteristics of cyclic group with the aim
of extending fundamental results and their implications within a broader context and applications to chemistry. We looked
at classical properties of cyclic groups, their generation, structure and subgroup behavior. We also explore generalizations
such as the decomposition of finite abelian into cyclic subgroup and their behavior under direct product constructions, and
the role of cyclicity in automorphism groups and homomorphic images. Using proof-based analysis, we show that these
generalized properties reveal deeper structural insights and enable a quick understanding of algebraic systems. Applications
to chemistry are also discussed to highlight the practical relevance of cyclic group theory. The paper concludes the directions
to future research on cyclic groups in some complex algebraic systems.
