Permutations

Abstract

We continue to increase our stock of examples by introducing groups of permutations. Rearranging or permuting the members of a set is a familiar idea, for example interchanging 1 and 3 while leaving 2 fixed gives a permutation of the first three integers. By a permutation of an arbitrary set X we shall mean a bijection from X to itself. The collection of all permutations of X forms a group Sx under composition of functions. There is very little to check. If α: X → X and β: X → X are permutations, the composite function αβ: X → X defined by αβ(x) = α(β(x)) is also a permutation. Composition of functions is associative, and the special permutations which leaves every point of X fixed clearly acts as an identity. Finally, each permutation a is a bijection and therefore has an inverse α−1: X → X, which is also a permutation and which satisfies α−1 α = ε = αα−1. If Xis an infinite set, S x is an infinite group. When X consists of the first n positive integers, then S x is written Sn and called the symmetric group of degree n. The order of Sn is n!.