Is there a homomorphism $f$ from $(\Bbb Z_4, +_4)$ to $(\Bbb Z, +)$, such that $f(1) =1$?

Question

I think that there is no such homomorphism, at least I couldn't find it. Is it correct?

Answer 1

There is no such homomorphism, since we have the following:

\begin{equation*} 0= f(0) = f(1 + 1 + 1 + 1) = 4f(1) \implies f(1) = 0 \end{equation*}