Any group of 3 elements is isomorphic to Z3

Prove that any group with three elements is isomorphic to Z3

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

We need to show 1-1, onto, and the homomorphism property

I'm having problems with all of the properties

Assume c(a)=c(b)

This implies 1=2, so c(a) is not equal to c(b).

Does this show 1-1?

Onto:

Need to show y=c(x) and solve for x

y=c(a)=1

y=1

I'm confused on this

Homomorphism:

c(a)c(b)

1*2

2=c(ab)

I know this should equal c(ab), but I didn't see why 2=c(ab)