Abstract Algebra Theory and Applications - Computer Science ...

4.1 DEFINITIONS AND NOTATION 734.1 Definitions and NotationIn general, the permutations of a set X form a group S X . If X is a finiteset, we can assume X = {1, 2, . . . , n}. In this case we write S n instead ofS X . The following theorem says that S n is a group. We call this group thesymmetric group on n letters.Theorem 4.1 The symmetric group on n letters, S n , is a group with n!elements, where the binary operation is the composition of maps.Proof. The identity of S n is just the identity map that sends 1 to 1, 2 to2, . . ., n to n. If f : S n → S n is a permutation, then f −1 exists, since f isone-to-one and onto; hence, every permutation has an inverse. Compositionof maps is associative, which makes the group operation associative. Weleave the proof that |S n | = n! as an exercise.□A subgroup of S n is called a permutation group.Example 1. Consider the subgroup G of S 5 consisting of the identitypermutation id and the permutations( )1 2 3 4 5σ =1 2 3 5 4( )1 2 3 4 5τ =3 2 1 4 5( )1 2 3 4 5µ =.3 2 1 5 4The following table tells us how to multiply elements in the permutationgroup G.◦ id σ τ µid id σ τ µσ σ id µ ττ τ µ id σµ µ τ σ idRemark. Though it is natural to multiply elements in a group from left toright, functions are composed from right to left. Let σ and τ be permutationson a set X. To compose σ and τ as functions, we calculate (σ ◦ τ)(x) =σ(τ(x)). That is, we do τ first, then σ. There are several ways to approach