# Problem of the Week #288 - Apr 24, 2019

Status
Not open for further replies.

#### Euge

##### MHB Global Moderator
Staff member
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.

-----

#### Euge

##### MHB Global Moderator
Staff member
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 $1$.

Status
Not open for further replies.