Abelian Transitive Action is Regular

Theorem

Let $G$ be an abelian transitive subgroup of the symmetric group $S_n$. Show that $G$ has order $n$.

\begin{proof} For any $x\in \{1,\cdots ,n\}$, there exists $\sigma \in G$ such that $\sigma (1)=x$. Let $G_1$ be the point stabilizer of $1$. Then $G_x=G_1^{\sigma }=G_1$ by $G$ abelian.

If $G_1\ne \{1\}$, then there is a non-trivial $\tau \in G_1$ and so $\tau \in G_x$ for all $x\in \{1,\cdots ,n\}$, i.e. $\tau$ is trivial on $\{1,\cdots ,n\}$, contradiction. \end{proof}