Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Transcript

Chapter 6: Isomorphisms Definition and Examples Cayley’ Theorem Automorphisms How to prove G is isomorphic to Examples: Example 1: Let G ( R,) and G ( R ,.), then G and G are isomorphic . 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 : G G given by ( x) x is not an isomorphis m 3 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: Step2: Step3: Step4: indeed a function one to one onto 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 Then U (12) {T1 , T5 , T7 , T11} and the multiplication tables for both groups is given by: Proof: (Theorem 6.2) Example: * * Show that (C ,.) and (R ,.) are not isomorphic Solution : Look at the number of solutions of x 4 1 in both groups Proof: (Theorem 6.3) Automorphisms DEFINITION:AUTOMORPHISIM Example : M : SL(2 R) SL(2, R) given by M ( A) MAM 1 is an automorphi sm Example: : C C give n by (a bi) a bi is an automorphi sm of C. Inner automprphosms Example : M : SL(2, R) SL(2, R) given by M ( A) MAM is an inner automorphi sm 1 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, Aut ( Z10 ) U (10) Proof; continue