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Any group of 3 elements is isomorphic to Z3 
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#1
Nov110, 07:40 PM

P: 603

1. The problem statement, all variables and given/known data
Prove that any group with three elements is isomorphic to [tex]Z_{3}[/tex] 2. Relevant equations 3. The attempt at a solution Let G be the group of three elements We have an isomorphism if given c:G>[tex]Z_{3}[/tex], if c is oneto one and onto and c(ab)=c(a)c(b) First, we check onetoone We want c(a)=c(b) to imply a=b My problem here is how to define c(a), c(b). Onto: We want c(a)=x and want to solve for a? c(ab): Same problem with not knowing what c(ab) is 


#2
Nov110, 07:45 PM

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P: 18,210

Take a group of three elements {e,a,b}. Since the order of every element must be three, we have that b=aČ. Thus the group is {e,a,aČ}.
Define the map G > Z3 by e > 0 a > 1 b > 2 It is easily checked that this is indeed an iso. 


#3
Nov110, 07:50 PM

P: 603

So c(a)=c(b)
c(a)=1 c(b)=2 1=2 not true, but that means it's not an isomorphism 


#4
Nov110, 07:53 PM

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P: 18,210

Any group of 3 elements is isomorphic to Z3
Uh what? c(a) doesnt equal c(b)??? does it?



#5
Nov110, 07:59 PM

P: 603

I thought for the one to one part, you assume c(a)=c(b)



#7
Nov110, 08:06 PM

P: 603

but I assumed that but that amounts to 1=2. How does that work?



#8
Nov110, 08:15 PM

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P: 18,210

You assumed that c(a)=c(b), and from that assimption followed that 1=2. So your assumption is wrong, and thus [tex]c(a)\neq c(b) [/tex]



#9
Nov110, 08:33 PM

P: 603

Then I guess I don't see how to show 11
onto y=c(x) Do I just take any element, say a y=c(a)=1 y=1, but we want to solve for x I thought c(a)c(b) 1*2 2=c(ab) 


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