# The symmetric group

Before defining anything, we shall provide a little motivation for some general notions which will arise later.

Let be any finite set and let denote the set of all permutations of onto itself:

This set has the following properties:
1. if belong to then (the composition of these permutations) also belongs to (closed under compositions"),

2. if all belong to then (associativity"),
3. the identity permutation belongs to (existence of the identity"),
4. if belongs to then the inverse permutation also belongs to (existence of the inverse").

The set is called the symmetric group of . We shall usually take for the set a set of the form , in which case we shall denote the symmetric group by . Note that the elements of are exactly the permutations first introduced in the previous chapter. This group is also called the symmetric group on letters.

Example 5.3.1   Suppose . We can describe in disjoint cycle notation as

We can compute all possible products of two elements of the group and tabulate them:

Exercise 5.3.2   Verify the four properties of mentioned above for Example 5.3.1. (Note that the verification of associativity follows from the associative property of the composition of functions - see Exercise 4.1.15).

david joyner 2008-04-20