SECTION 8 Groups of Permutations Definition A permutation of a set A is a function  ϕ : A  A that is both one to one and onto. If  and  are both permutations of a set A, then the composite function   defined by gives a one-to-one and onto mapping of A into A. We can show that function composition  is a binary operation, and call this function composition  permutation multiplication. We will denote   by  . Remember that the action of   on A must be read in right-to-left order: first apply  and then .

Notations Example: Suppose A = {1, 2, 3, 4, 5} and that  is the permutation given by 1  4, 2  2, 3  5, 4  3, 5  1. We can write  as following: Let, then  =

Permutation Groups Theorem Le A be a nonempty set, and let S A be the collection of all permutations of A. Then S A is a group under permutation multiplication. Proof: exercise.

Symmetric Groups Note: here we will focus on the case where A is finite. it’s also customary to take A to be set of the form {1, 2, 3, …,, n} for some positive integer n. Definition: Let A be the finite set {1, 2,   , n}. The group of all permutations of A is the symmetric group on n letters, and is denoted by S n. Note that S n has n! elements, where n!=n(n-1)(n-2)    (3)(2)(1).

Two important examples Example: S 3 Let set A be {1, 2, 3}. Then S 3 is a group with 3!=6 elements. Let Then the multiplication table for S 3 is shown in the next slide.

S 3 and D 3 Note that this group is not abelian ( ) There is a natural correspondence between the elements of S 3 and the ways in which two copies of an equilateral triangle with vertices 1, 2, and 3 can be placed, one covering the other with vertices on to of vertices. For this reason, S 3 is also the group D 3 of symmetries of an equilateral triangle. Naively, we used  i for rotations and  i for mirror images in bisectors of angles. 3 12

Cayley’s Theorem Definition Let f: A  B be a function and let H be a subset of A. The image of H under f is { f (h) | h  H } and is denoted by f [H]. Lemma Let G and G’ be groups and let  : G  G’ be a one-to-one function such that  (x y) =  (x)  (y) for all x, y  G. Then  [G] is a subgroup of G’ and  provides an isomorphism of G with  [G]. Then apply the above Lemma, we can show Theorem (Cayley’s Theorem) Every group is isomorphic to a group of permutations.