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?

Alternatively, you could try to calculate the kernel of the homomorphism $\mathbb{Z}\oplus\mathbb{Z}\rightarrow \mathbb{Z}/12$ given by $(x,y)\mapsto 4x+3y$. With any luck, the kernel will turn out to be $3\mathbb{Z}\oplus 4\mathbb{Z}$.
Then apply the first isomorphism theorem to say that $\mathbb{Z}/12\cong \mathbb{Z}\oplus \mathbb{Z}/(3\mathbb{Z}\oplus 4\mathbb{Z})$.
