Show that $(U_{7}, \cdot) \cong (Z_6, +)$.

Question

Show that $(U_{7}, \times) \cong (Z_6, +)$. ($U_{7}$ is the set of units in $Z_7$, and $Z_6$ is the set of integers modulo 6)

I understand this in theory, that we want to define a function $\phi : U_{7} \rightarrow Z_{6}$ that's a homomorphism, and is one-to-one and onto. But there's a lot of things about this I feel like I don't understand or that haven't been explained to me fully.

How do you decide how to define the function?

Is it necessary to prove separately that the function is a homomorphism, and that it's one-to-one, and that it's onto?

How do you prove that a function is a homomorphism? How do you prove a function is onto?

Edit: Following HallaSurvivor's advice I figured out a function that I think should work: $\phi([x]_6)=[3^x]_7$

So now I'm left with my other questions: How do I prove this is a homomorphism and that this is onto, and is it even necessary to do so?

Answer 1

$\textbf{Hint}$ : Think of a map $f$ : $\mathbb{Z}_6$ $\rightarrow$ $U_7$ sending a $generator$ to a $generator$ of respective group. Rest is algebraic manipulation.