The Symmetric Group, denoted as S_n , is a mathematical concept in group theory that represents the set of all possible permutations of n elements. Each permutation is a rearrangement of these elements, and the group operation is the composition of these permutations. The size of the symmetric group S_n is n! (n factorial), which counts the total number of ways to arrange n distinct objects. Symmetric groups are fundamental in various areas of mathematics, including algebra, combinatorics, and geometry. They help in understanding the structure of more complex groups and are used in solving problems related to symmetry and transformations. The study of symmetric groups also leads to important concepts like group homomorphisms and group actions.