Showing that $\mathbb{Z}_3\times \mathbb{Z}_4$ is Isomorphic to $\mathbb{Z}_{12}$ How can I show that  $\mathbb{Z}_3\times\mathbb{Z}_4$ is isomorphic to $\mathbb{Z}_{12}$ I found an order twelve generator, and that was the hint, but can I show it is an isomorphism without showing it's a bijective homomorphism, or is that the only way?

Just define $f : \mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z} \to \mathbb{Z}/12\mathbb{Z}$ by
$$
f([a],[b]) = [ab]
$$
You need to check that this is well-defined (ie. if $a \equiv c\pmod{3}$ and $b\equiv d\pmod{4}$, then $ab\equiv cd\mod{12}$). This takes a little work, but once that is done, it is clear that it is the isomorphism you are looking for.
