1. Chapter 6: Isomorphisms
Definition and Examples
Cayley’ Theorem
Automorphisms

5. How to prove G is isomorphic to

6. Examples:
Example 1:

7. Example 2: Let G=<a> be an infinite cyclic group
Example 2: Let G=<a> be an infinite cyclic group. Then G is isomorphic to Z.

8. Example 3: Any finite cyclic group of order n is isomorphic to Z_n.

9. Example 4: Let G=(R,+). Then

10. Example 5:

11. Example 6:
U(12)={1,5,7,11} 1.1=1, 5.5=1, 7.7=1, 11.11=1 That is x^2=1 for all x in U(12)

12. Example 7:

13. Example 8: Step1: indeed a function Step2: one to one Step3: onto
Step4: preserves multiplication

14. Caylay’s Theorem
Theorem 6.1: Every group is isomorphic to a group of permutations.

15. Example:
Find a group of permutations that is isomorphic to the group U(12)={1,5,7,11}. Solution: Let
and the multiplication tables for both groups is given by:

19. Proof: (Theorem 6.2)

20. Example:

22. Proof: (Theorem 6.3)

23. Automorphisms

24. Definition:Automorphisim

25. Example:

26. Inner automprphosms

27. What are the inner automorphisms of D_4?

28. Definition:

30. Inn(G)

32. Determine all automorphisms of Z_10
That is, find Aut(Z_10). Show that Aut(Z_10) is a cyclic group. Moreover,

35. Proof; continue