1. Chapter 8: External Direct Product
Definition and Examples
Properties of External Direct Product
The group U(n) as an external direct product

2. Definition and Examples

3. Examples

4. Examples;

5. Examples;

7. Proof: Theorem 8.1

8. Examples

9. Example;

10. Solution

11. Solution; cont

12. Example

16. Example

18. Recall that for k divides n,
For example

20. Corollary

21. Example U(105)

22. U(n) is isomorphic to a product of cyclic group

23. Example; U(105), U(720)

24. U(720)
We may ask the following questions:
What is the order of U(720)?
Is U(720) cyclic?
What are the orders of elements in U(720)?
How many elements of order 12 are there in U(720)?
Since Aut(Z_n) is isomorphic to U(n), all these questions are valid for Aut(Z_n)