# Problem of the Week #237 - Dec 13, 2016

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#### Euge

##### MHB Global Moderator
Staff member
Here is this week's POTW:

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

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#### Euge

##### MHB Global Moderator
Staff member
This week's problem was solved by johng . You can read his solution below.

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 $[G:Z]\geq[G:C_G(x)]$.

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