The Significance of Cyclic Groups

Abstract

Abstract Investigation in group theory has been a widespread practice within the realm of mathematics. In finite group theory specifically, mathematicians have historically aimed to classify finite simple groups. Following the turning consensus that the finite simple group classification is complete, research within finite group theory has shifted to other areas. However, this paper endeavors to fill the missing gaps in the current classification – it does not specify the importance of finite group theory. By focusing on introducing, explaining, and demonstrating cyclic groups and their applications, the paper will reveal, specifically, the importance of cyclic groups in the realm of mathematics and beyond. In particular, the qualities of finite cyclic groups allow them to apply easily and broadly from as simple as division rules and binary number systems (by use of a set of integers of modulus n), to more advanced theory and implementation with computer science. Cyclic group theory deserves increased recognition. In addition, by providing definitions, evaluations, proofs, and concrete examples, this paper also endeavors to become a point of reference for readers to understand, consolidate, and further grasp cyclic group theory and its uses.