
Originally Posted by
kathrynmath
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)