# Any group of 3 elements is isomorphic to Z3

by kathrynag
Tags: elements, isomorphic
 P: 607 1. The problem statement, all variables and given/known data Prove that any group with three elements is isomorphic to $$Z_{3}$$ 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--->$$Z_{3}$$, if c is one-to -one and onto and c(ab)=c(a)c(b) First, we check one-to-one 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
 PF Patron Sci Advisor Thanks Emeritus P: 15,673 Take a group of three elements {e,a,b}. Since the order of every element must be three, we have that b=a². Thus the group is {e,a,a²}. Define the map G --> Z3 by e ---> 0 a ---> 1 b ---> 2 It is easily checked that this is indeed an iso.
 P: 607 So c(a)=c(b) c(a)=1 c(b)=2 1=2 not true, but that means it's not an isomorphism
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## Any group of 3 elements is isomorphic to Z3

Uh what? c(a) doesnt equal c(b)??? does it?
 P: 607 I thought for the one to one part, you assume c(a)=c(b)
 PF Patron Sci Advisor Thanks Emeritus P: 15,673 Yes... never mind...
 P: 607 but I assumed that but that amounts to 1=2. How does that work?
 PF Patron Sci Advisor Thanks Emeritus P: 15,673 You assumed that c(a)=c(b), and from that assimption followed that 1=2. So your assumption is wrong, and thus $$c(a)\neq c(b)$$
 P: 607 Then I guess I don't see how to show 1-1 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)
 PF Patron Sci Advisor Thanks Emeritus P: 15,673 You show 1-1 by a simple proof by contradiction. I'm sorry, but a math major really should be able to do such a thing...

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