How to construct an isomorphism?

Question

I know the group $\mathbb{Z}_3 \times \mathbb{Z}_3 \times \mathbb{Z}_2$ is isomorphic to $\mathbb{Z}_3 \times \mathbb{Z}_6$ (since $\mathbb{Z}_3 \times \mathbb{Z}_2$ is isomorphic to $\mathbb{Z}_6$ since $2$ and $3$ are coprimes).

But how would I construct such an ismorphism?

Answer 1

You can make it explicit by projecting the elements of $Z/6$ to the respective elements in $Z/3$ and $Z/2$, i.e. $$Z/6 \rightarrow Z/3 \times Z/2, i \mapsto (i \text{ mod } 3, i \text{ mod }2).$$ As you say, 2 and 3 are coprime and thus the Chinese remainder theorem tells you that this is an isomorphism.

Answer 2

An isomorphism $f: \mathbb{Z_3 \times Z_2} \to \mathbb{Z_6}$ can be determined by $f(1,1) \to 1$

So it would make sense that the map $\mathbb{Z_3 \times Z_3 \times Z_2} \to \mathbb{Z_3 \times Z_6}$ could be given by

$$(0,a,b) \mapsto (0,f(a,b)) \\(1,a,b) \mapsto (1,f(a,b))\\(2,a,b) \mapsto(2,f(a,b)) $$