IMC 2010 · Problem 3

Denote by $S_n$ the group of permutations of the sequence $(1, 2, \dots, n)$. Suppose that $G$ is a subgroup of $S_n$, such that for every $\pi \in G \setminus \{e\}$ there exists a unique $k \in \{1, 2, \dots, n\}$ for which $\pi(k) = k$. (Here $e$ is the unit element in the group $S_n$.) Show that this $k$ is the same for all $\pi \in G \setminus \{e\}$.