If $G = \langle A, N \rangle$ ... , show that $Z(G)$ has finite index.

Question

Trying to solve the following problem in preparation for pre-lims! Thanks in advance for any help!

Let $G$ be a group. Assume that $G$ is generated by two subgroups, $N$ and $A$, such that $A$ is abelian and $N$ is finite and normal in $G$. Show that the center $Z(G)$ has finite index in $G$.

We note that if $G$ is finite or abelian, the result falls out easily. We are stuck on the infinite, non-abelian case.

Answer 1

$N$ finite implies that $|G:C_G(N)|$ is finite. Also $G=AN \Rightarrow |G:A|$ finite. So $|G:C_A(N)|$ is finite, and $C_A(N) \le Z(G)$.