analytic functions – Homomorphism and holomorphic funtions

Let $M leq mathbb{C}$ the aditive subgroup such that, $$M={n+im in mathbb{C}:n,min mathbb{Z} }$$ Let $f$ be a entire funtion such that $f(0)=0$, suppose that $$g(z+M)=f(z)+M, (zinmathbb{C})$$
define a group homomorphism, $g:mathbb{C}/M rightarrowmathbb{C}/M$, where $mathbb{C}/M$ is the quotient group.

$(a)$ Prove that exists some disk $D$ centered in $0$, such that $f(z+w)=f(z)+f(w)$, for all $z,w in D.$

$(b)$ Prove that exists some $binmathbb{C}$, such that $f(z)=az$, for all $z$ in some neighborhood of $0$.

$(c)$ Prove that exists some $bin mathbb{C}$, such that $g(z+M)=az+M$ for all $z in mathbb{C}.$

I don’t se a clear path, because we never use group properties in the complex variable course, however, as $f$ is entire and $f(0)=0$, there exist an holomorphic funtion $h$, such that $f(z)=z^rh(z)$ for some $rgeq1$, and $h(0)neq0$, but how can I use this?

Even so is not clear for me, how to use the quotient, of wich propertie is important in the Gaussian integers in this case.