Welcome to our community

Be a part of something great, join today!

Problem of the Week #237 - Dec 13, 2016

Status
Not open for further replies.
  • Thread starter
  • Moderator
  • #1

Euge

MHB Global Moderator
Staff member
Jun 20, 2014
1,892
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 POTW submission guidelines to find out how to submit your answers!
 
  • Thread starter
  • Moderator
  • #2

Euge

MHB Global Moderator
Staff member
Jun 20, 2014
1,892
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 $[G:Z]\geq[G:C_G(x)]$.
 
Status
Not open for further replies.