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

[Euge]

Here is this week's POTW:

-----
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.

-----

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!

 

[Euge]

This week's problem was solved correctly by castor28 and Olinguito. You can read castor28's solution below.

Spoiler

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 $1$.