Chapter 8: External Direct Product

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Presentation transcript:

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

Definition and Examples

Examples

Examples;

Examples;

Proof: Theorem 8.1

Examples

Example;

Solution

Solution; cont

Example

Example

Recall that for k divides n, For example

Corollary

Example U(105)

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

Example; U(105), U(720)

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)