Show that any abelian transitive subgroup of $S_n$ has order $n$

Question

Can anybody tell me what is known about the classification of abelian transitive groups of the symmetric groups?

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

Thanks for your help!

Answer 1

Reading this question, an alternative solution came to my mind. It is shorter than my original solution, but slightly less elementary as it uses (very basic) theory of group actions.

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By transitivity, the orbit of $1$ is $G\cdot 1 = \{1,\ldots,n\}$.

Let $\sigma\in G_1$, where $G_1 = \{\sigma\in G \mid \sigma(1) = 1\}$ is the stabilizer of $1$. Let $x\in \{1,\ldots,n\}$. Transitivity gives a $\tau\in G$ with $\tau(1) = x$. Now $$\sigma(x) = \sigma\tau(1) \overset{G\text{ abelian}}{=} \tau\sigma(1) = \tau(1) = x,$$ showing that $\sigma = \operatorname{id}$ and therefore $G_1 = \{\operatorname{id}\}$.

Now by the orbit-stabilizer theorem, $$\#G = \#(G\cdot 1) \cdot \#G_1 = n\cdot 1 = n.$$