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
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 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
Example:
isomorphicnot are ,.)(R and ,.)( that Show **C
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DEFINITION:AUTOMORPHISIM
smautomorphian is )(by given ),2()2(:
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What are the inner automorphisms of D_4?
Determine all automorphisms of Z_10That is, find Aut(Z_10). Show that Aut(Z_10) is a cyclic group. Moreover,
)10()( 10 UZAut