A symmetric group is a mathematical concept in the field of group theory, which studies the algebraic structures known as groups. Specifically, the symmetric group on a set of n elements, denoted as S_n, consists of all possible permutations of those elements. The group operation is the composition of these permutations, meaning that applying one permutation after another results in another permutation. Symmetric groups are important in various areas of mathematics, including combinatorics and algebra. They help in understanding the symmetries of objects and play a crucial role in Galois theory, which connects field theory and group theory. The size of the symmetric group S_n is n! (n factorial), representing the total number of ways to arrange n elements.