1. ## isomorphic

Let $G=\mathbb{Z}_4\times\mathbb{Z}_3$. Show that $G$ is isomorphic to $\mathbb{Z}_{12}$.

2. Take $(1,1)\in \mathbb{Z}_4 \times \mathbb{Z}_3$ then let $k\in \mathbb{N}$ and we have $(k1,k1)=0$ iff $4\vert k$ and $3\vert k$ the least such number is $[4,3]$ (the min. comm. mult.) and it's well known that $[a,b](a,b)=ab$ then since $(4,3)=1$...

3. ## One to one and onto

The map
Z12 -> G
defined by n -> (n mod 4, n mod 3 ) is 1:1 and onto.

Alternatively,
G-> Z12
defined by
(a,b) -> 3*a + 4*b mod 12
should also be 1:1 and onto.