The symmetric group is a mathematical concept that represents all the possible ways to arrange a set of objects. For a set with n elements, the symmetric group is denoted as S_n and contains n! (n factorial) permutations. Each permutation is a unique arrangement of the elements in the set. Symmetric groups are fundamental in the field of group theory, a branch of abstract algebra. They help mathematicians understand the structure of symmetries and transformations in various mathematical contexts, including geometry and combinatorics.