My question is taken from exercise IV.3.4(b) in Brown's group cohomology book. Let $E$ be a finitely generated group, and suppose that $C$, the center of $E$, has finite index in $E$, say $[E:C]=n$. The exercise asks us to show that the commutator subgroup $[E,E]$ is finite.
From part $(a)$ of the same exercise, we have a homomorphism $E\to C$ which is the $n$th power map when restricted to $C$, and the hint given in the book says to notice that this map has a finite kernel. I'm having trouble seeing this fact.
If it is helpful, the map $E\to C$ is obtained as follows. Consider the extension
$$1\to C\to E\to E/C=G\to 1$$
This represents an element $\alpha$ in $H^2(G,C)$, and if $f:C\to C$ is the $n$th power map, then $H^2(G,f)(\alpha)=0\in H^2(G,C)$ (this is because $|G|=n$ annihilates cohomology groups). It follows that there is a map from $E$ to the trivial extension $C\times G$, whose first component is a homomorphism $E\to C$ which restricts to the $n$th power map on $C$ (the extensions must fit into a commutative diagram with $f$).
Hints or answers are much appreciated. Thanks!