The structure theorem for finite abelian group is one of the fundamental results in modern algebra, providing a complete classification of such groups as direct products of cyclic groups of prime power order. This decomposition not only simplifies the study of group – theoretic properties but also offer powerful tools for practical applications in various domains of mathematics and computer science. In this paper, we revisit the decomposition theorem of finite abelian groups, present illustrative examples for groups of small order, and highlight its applications in number theory, cryptography, and coding theory. In particular, we discuss how the theorem aids in understanding the structure of cyclic subgroups, facilitates efficient computations in modular arithmetic, and supports the construction of secure cryptographic protocols. Furthermore, the paper explores connections of the decomposition with the classification of finite modules over principal ideal domains (PIDs). The study underscores the significance of the decomposition theorem as both a theoretical cornerstone and a practical instrument in contemporary mathematical research.
Mr. Abhishek Bisai (Wed,) studied this question.