Lecture 21: Remarks on permutations and determinants 1. Permutations, Existence and uniqueness of the determinant Summary: Lecture 21: Remarks on permutations and determinants 1. Permutations, Existence and uniqueness of the determinant A permutation of the set {1, . . ., n} is a rearrangement of the sequence (1, . . . , n). The following list contains all permutations of the set {1, 2, 3}. There are 3! = 6 of them (1, 2, 3) , (1, 3, 2) , (2, 1, 3) , (2, 3, 1) , (3, 1, 2) , (3, 2, 1). The formally correct definition is the following: A permutation of the set {1, . . ., n} is a function : {1, . . . , n} - {1, . . ., n} such that j = k implies (j) = (k). For example, the permutation (1, 3, 2) in the above list corresponds to the function (1) = 1 , (2) = 3 , (3) = 2. Some permutations simply switch two elements of the sequence, such as the per- mutation (3, 2, 1). We will call them 'simple permutations'. If ((1), . . . , (n)) is a permutation, and if N simple permutations applied successively transform it back to (1, . . . , n) then (-1)N is called the sign of the permutation . Here are the signs of the permutations of the set {1, 2, 3} permutation sign (1, 2, 3) +1 Collections: Mathematics