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.

Nesting IF AND funtions in Google Sheets

I have 3 drop down boxes that I want one cell to choose an answer for based on the selections for each drop down. I can use the IF AND function to produce the result I want. Here is that formula: =if((AND(A13=”Picasso 1000″,C13=”RIC”,E13=”Batteries”)),”$4,392″,””). I either can’t figure out how to properly nest several of these types of formulas for other outcomes or I am not using the right kind of formula to produce the result I need. Thank in advance!