If a subgroup $H\le S_n$ acts transitively on transpositions then $H=S_n$

Question

Let $H \le S_n$ acts on the set of transpositions:

$X=\{(i \space j)\space | \space i\ne j\}\subseteq S_n$

by conjugation and assume the action is transitive.

Assume $(1 \space 2)\in H$. Prove: $H=S_n$

My try:

I know from the Orbit-Stabilizer theorem that $|X|=\frac{n\cdot(n-1)}{2}$ divides $|H|$.

I know that the transpositions generate $S_n$. So I was thinking maybe I could show that all transpositions are in $H$.

Will appreciate any hint or direction.

Answer 1

Yes, you can show that all transpositions are in $H$. You have $(1 \space 2) \in H$, and every $(i \space j)$ is of the form $h (1 \space 2) h^{-1}$ for some $h \in H$. Hence $(i \space j) \in H$.