I have been looking at J Neukirch's book on Algebraic Number Theory, he has a chapter on Abstract class field theory, in which he talks about Abstract valuation theory, this is where ive run into a problem. This is the set up.
We let $G$ be a profinite group, and assume we have a surjective continuous homomorphism $d:G \rightarrow \hat{\mathbb{Z}}$, now we let $G_{k}$ be a subgroup of $G$, so we can then restrict $d$ to $G_{k}$ and get a homomorphism $d:G_{k} \rightarrow \hat{\mathbb{Z}}$, he then defines $f_{k} = (\hat{\mathbb{Z}}:d(G_{k}))$.
Now what I dont understand is the following, he says if $f_{k}$ is finite then we get a surjective homomorphism $d_{k}=\frac{1}{f_{k}}d:G_{k} \rightarrow \hat{\mathbb{Z}}$, why is this surjective? and is there a good way to visualize this map when its applied to the absolute galois groups of local or global fields?
Thank you