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

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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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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 $1$.