Determine all group homomorphisms of the form $\phi: \mathbb{Z}_{12}\to \mathbb{Z}_{30}$.

Question

I have a quick question about the following problem:

Determine all group homomorphisms of the form $\phi: \mathbb{Z}_{12}\to \mathbb{Z}_{30}$.

I understand that $\phi(1)$ generates $\phi$, and since $|\phi(1)|$ divides both $12$ and $30$, then $|\phi(1)|=\left \{ 1,2,3,6 \right \}$. This is where I get confused: $|\phi(1)|=\left \{ 1,2,3,6 \right \}$ implies $\phi(1)=\left \{ 0,15,10,20,25 \right \}$ which are all in $\mathbb{Z}_{30}$. I don't understand this part...where did $0,15,10,20,25$ come from?

Answer 1

The homomorphism $\phi:\mathbb{Z}_{12}\rightarrow\mathbb{Z}_{30}$ is determined by $\phi(1)$.

$\phi(12)=12\cdot\phi(1)\equiv_{30}0$, so you need to find all the $\phi(1)\in\mathbb{Z}_{30}$ such that $30\mid 12\cdot \phi(1)$.

Answer 2

Hint: Which elements $x \in \Bbb Z_{30}$ satisfy $6x = 0$?