MHB Can the center of a finite group determine the size of its conjugacy classes?

[Euge]

Here is this week's POTW:

-----
Let $Z$ be the center of a finite group $G$. Prove that there are at most $(G : Z)$ elements in each conjugacy class of $G$.

-----

Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!

 

[Euge]

This week's problem was solved by johng. You can read his solution below.

Spoiler

Let $x\in G$. The number of conjugates of $x$ (the cardinality of the class of $x$) is $[G:C_G(x)]$ where $[G:C_G(x)]=\{g\in G\,:\,gx=xg\}$. Since obviously $Z\subseteq C_G(x),\,\,[G:Z]=[G:C_G(x)][C_G(x):Z]$ and so $[G:Z]\geq[G:C_G(x)]$.