Let X = {1, 2, ..., n}. We introduce permutations and describe multiplication (taking products) by menas of composition of maps.

Definition

A bijection of X to itself is also called a permutation. The set of all permutations of X is denoted by Sn.

The product of two permutations g, h in Sn is defined as the composition of g and h. Thus, for all x X,

gh(x) = g(h(x)).



The product of two permutations in Sn is again a permutation and hence an element of Sn. (Prove this!)

The identity map id from X to X plays a special role: id o g = g o id = g, for all g Sn. The inverse of g Sn, denoted by g-1, is again a permutation and satisfies g o g-1 = g-1 o g = id. We say: id is the identity element for the product on Sn. We often use e to denote the identity element. For every positive integer m, we denote by gm the product of m factors g. Instead of (g-1)m we write g-m.

We call Sn the symmetric group of degree n. The symmetric group is an instance of the structure group that will be discussed in Chapter 6.