Chapter 6: Isomorphisms

Definition and Examples

Cayley’ Theorem

Automorphisms

How to prove G is isomorphic to

Examples: Example 1:

.isomorphic are and then ,.),( and ),(Let GGRGRG

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 isisomorphic to Z_n.

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

misomorphisan not is )(by given : 3xxGG

Example 5:

U(12)={1,5,7,11}1.1=1, 5.5=1, 7.7=1, 11.11=1That is x^2=1 for all x in U(12)

Example 6:

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:

},,,{)12(Then 11751 TTTTU

Proof: (Theorem 6.2)

Example:

isomorphicnot are ,.)(R and ,.)( that Show **C

groupsboth in 1 of solutions ofnumber at theLook

:Solution4 x

Proof: (Theorem 6.3)

Automorphisms

DEFINITION:AUTOMORPHISIM

smautomorphian is )(by given ),2()2(:

:Example1 MAMARSLRSL MM

Example:

. of smautomorphian is

)(by n give :

C

biabiaCC

Inner automprphosms

smautomorphiinner an is

)(by given ),2(),2(:

:Example1 MAMARSLRSL MM

What are the inner automorphisms of D_4?

Definition:

Inn(G)

Determine all automorphisms of Z_10That is, find Aut(Z_10). Show that Aut(Z_10) is a cyclic group. Moreover,

)10()( 10 UZAut

Proof; continue