Center of a group with finite index

[mahler1]

Homework Statement

Let $G$ be a group such that its center $Z(G)$ has finite index. Prove that every conjugacy class has finite elements.

Homework Equations

The Attempt at a Solution

I know that $[G:Z(G)]<\infty$. If I consider the action on $G$ on itself by conjugation, each conjugacy class is identified with an orbit, and for each orbit $\mathcal0_x \cong G/C_G(x)$, where $C_G(x)$ is the stabilizer of $x$ by the action, in this particular case, the centralizer of $x$. I got stuck here, I know that $Z(G) \leq C_G(x)$ for all $x \in G$, I don't know how to deduce from here that $[G:C_G(x)]<\infty$, I would appreciate some help.

 

[pasmith]

Hint: Show that if $g_1$ and $g_2$ are in the same coset of $Z(G)$ then for all $h \in G$ we have $g_1hg_1^{-1}= g_2hg_2^{-1}$.

(Edit: the converse also holds, but I don't think you will require that.)

Given any $h \in G$ you can then show that $\phi_h : G/Z(G) \to [h] : gZ(G) \mapsto ghg^{-1}$ is a surjection, where $[h]$ is the conjugacy class of $h$. (The first hint allows us to define such a function.)