1. Cayley Theorem Every group is isomorphic to a permutation group.

2. Example: U(10) U(10) = {1, 3, 7, 9} Definition: For g in U(10), let T g (x)= gx T 1 (x) = T 3 (x) = T 7 (x) = T 9 (x) = x T 1 =  3x T 3 = (1 3 9 7) 7x T 7 = (1 7 9 3) 9x T 9 =(1 9)(3 7)

3. Every group is isomorphic to a permutation group Proof: Let G be any group. 1.For g in G, define T g (x) = gx. We show T g is a permutation on G. 2.Let S = {T g | for g in G} We show S is a permutation group. 3.Define the map  :G  S by  (g)=T g We show  is an isomorphism.

4. 1. T g is a permutation on G. Suppose T g (x) = T g (y). Then gx = gy. By left cancellation, x=y. Hence T g is 1 to 1. Choose any y in G. Let x = g -1 y Then T g (x) = gx = gg -1 y = y So T g is onto. This shows that T g is a permutation.

5. 2. {T g | g in G} is a group The operation is composition. For a,b,x, T a T b (x) = T a (bx) = a(bx) = (ab)x =T ab (x) So T a T b = T ab (*) From (*), T e T a = T ea = T a, So T e is the identity in S. If b = a -1 we have, T a T b = T ab = T e So T a -1 = T b and S has inverses. Function composition is associative. Therefore, S is a group.

6. 3.  (g) = T g is isomorphism 1.Choose a, b in G. Suppose  (a) =  (b). Then T a = T b. In particular, for any x in G, T a (x) = T b (x) ax = bx a = b Therefore  is one-to-one.

7.  (g) = T g is isomorphism 2.Choose any T g in S. Then  (g) = T g Therefore,  is onto.

8.  (g) = T g is isomorphism 3.Choose any a, b in G. Then  (ab) = T ab = T a T b by (*) =  (a)  (b) Therefore,  is Operation Preserving. It follows that  is an isomorphism.

9. Why do we care? The permutation group we constructed is called the Left Regular Representation of G. Every abstract group can be represented in a concrete way It shows that abstract groups are all permutation groups, unifying the study of both.