Prove that Z/12Z is not isomorphic to Z/4Z × Z/6Z

Question

I understand that Z/12 is not isomorphic to z/4z x z/z6z because 4 and 6 are not relatively prime. But I do not know how to prove that and I have no intuitive understanding of why they need to be relatively prime.

I know z/4z x z/z6z has 24 elements while z/12z only has 12 which could a be useful to show that the function is not "onto." But then again the 13th element in z/4z x z/z6z is simply the same as its 1st element so there is no element in z/4z x z/z6z which does not correspond to an element in z/12. Please advice.

Answer 1

If there are 12 elements in the first group and 24 in the other, how could there exist a bijection?

Answer 2

An isomorphism between groups is a bijective homomorphism. Without even stopping to consider the homomorphism part, a prerequisite condition for finite sets (as mentioned by John above) is that they must be of the same cardinality/size. In this case, $\mathbb{Z}/12\mathbb{Z}$ has $12$ elements, while $\mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}/6\mathbb{Z}$ has $24$ elements. A function mapping one to the other can not possibly be both injective and surjective, hence no isomorphism exists.

Answer 3

Here's an answer if you meant $Z/24Z$. That's where you need the fact that 4 and 6 aren't relatively prime. For $Z/12Z$ the other answers are just fine - the sets aren't even the same size.

Now $Z/24Z$ is cyclic of order 24, while $Z/4Z \times Z/6Z$ doesn't have an element of order 24. (Can you prove that?)