Can a Finite Group Acting on a Set Have No Fixed Points?

  • MHB
  • Thread starter Euge
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  • #1
Euge
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Here is this week's POTW:

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Let $X$ be a finite set with more than one element, and $G$ be a finite group acting transitively on $X$. Show that some element of $G$ is free of fixed points.

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Remember to read the https://mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to https://mathhelpboards.com/forms.php?do=form&fid=2!
 
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  • #2
This week's problem was solved correctly by castor28 and Olinguito. You can read castor28's solution below.
Let $|X|=n>1$. As the action is transitive, there is only one orbit. By Burnside's lemma, this is equal to the average number of points fixed by an element of $G$.
The identity of $G$ fixes all the $n$ points. If every element of $G$ fixed at least one point, the average would be grater than $1$.
 

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