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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