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Chapter 6: Isomorphisms

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

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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:

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19 Proof: (Theorem 6.2)

20 Example:

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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:

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30 Inn(G)

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

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35 Proof; continue


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