I always get a little bit confused when dealing with homomorphism questions and how to identify which binary operations needs to be used, for example I had the following question as homework
Is the following a homomorphism:
$\psi\colon \mathbb{C}^{\times}\rightarrow \mathbb{R}$
$\psi (a+bi)=b$ (for $a,b \in \mathbb{R}$)
I tried:
$\psi((a+bi)(c+di))= \psi((ac-bd)+i(bc+ad)) =bc+ad$
Then $\psi(a+bi)+\psi(c+di) =b+d$
Which would imply that it isn't a homomorphism however my teacher has informed me it is, have I used the wrong binary operation, and how do I know which one to use?
You are right that the binary operations that turn the sets in question into group are essential to the solution. But in fact they are already essential to the problem statement. If we talk about groups, however, it is often clear which group operatin should be considered (unless explicitly specified differently). That is, for $\mathbb Z$,$\mathbb Q$, $\mathbb R$, $\mathbb C$ one may safely assume that addition is the intended binary operation (and surely not multiplication as $0$ does not have an inverse); and with $\mathbb Q^\times$, $\mathbb R^\times$ (or $\mathbb R_{>0}$) or $\mathbb C^\times$ one may safely assume that multiplication is intended (and not addition as that would be lacking the neutral element). So in cour case with a map $\mathbb C^\times\to\mathbb R$, your assumptions were fully justified - and correctly lead to the conclusion that $\psi$ is not a group homomorphism.