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 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
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.
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]
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)
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...