Thread: Ring Homomorphism

I was asked to decide if Z_9 and the direct sum of Z_3 and Z_3 are isomorphic.

My initial thought is that they aren't isomorphic, but am unsure on how to justify this.

Originally Posted by mathematic

I was asked to decide if Z_9 and the direct sum of Z_3 and Z_3 are isomorphic.

My initial thought is that they aren't isomorphic, but am unsure on how to justify this.

For example, count the numer of elements of order 9 in both groups...

Tonio

Well for Z_9:
We have elements 0,1,2,3,4,5,6,7,8
For the direct sum:
I started by calculating some direct sums
(0,0)(0,0)=(0,0)
(0,0)+(0,0)=(0,0)
(0,0)(0,1)=(0,0)
(0,0)+(0,1)=(0,1)
(0,0)(0,2)=(0,0)
(0,0)+(0,2)=(0,2)
(0,0)(1,0)=(0,0)
(0,0)+(1,0)=(1,0)
(0,0)(2,0)=(0,0)
(0,0)+(2,0)=(2,0)
.....

Originally Posted by mathematic

I was asked to decide if Z_9 and the direct sum of Z_3 and Z_3 are isomorphic.

My initial thought is that they aren't isomorphic, but am unsure on how to justify this.

They aren't even isomorphic as groups. This follows from the fact that every element of has order three and thus can't be generated by any element and thus is not cyclic.