Chapter 6: Isomorphisms Definition and Examples Cayley’ Theorem Automorphisms

How to prove G is isomorphic to

Examples: Example 1:

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.

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

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

Example 5:

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)

Example 7:

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

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

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:

Proof: (Theorem 6.2)

Example:

Proof: (Theorem 6.3)

Automorphisms

Definition:Automorphisim

Example:

Inner automprphosms

What are the inner automorphisms of D_4?

Definition:

Inn(G)

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

Proof; continue