symmetric group on n letters

The symmetric group on n letters, usually written as Sₙ, is the collection of all possible ways to rearrange n different objects. Each element of Sₙ is a specific arrangement, or permutation, of these objects. For example, if n=3 and the objects are A, B, and C, then one permutation could be swapping A and C while keeping B fixed. The group structure comes from combining these permutations: doing one after another. These groups are fundamental in studying symmetry and structure in mathematics, as they describe all possible symmetries of a set with n elements.