Symmetric groups

In this paper, we explore applications, examples, and representative theo-ries of Symmetric groups in a symmetric group. All elements are all bijections to the set itself, and the group operation is function composition. We begin by discussing the definition of symmetry with a few basic examples and applications. We then introduce and define some real-world applications followed by properties and special elements of symmetric groups. Lastly, we show the subgroup structure of symmetric groups and some of the representative theories.