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.