Showing that a group with transposition acting transitively on ${1, \dots, n}$ is $S_n$ I am trying to prove that a group is $S_n$ where $n$ is any integer.
Right now I have gotten that it acts transitively on $\{1, \dots, n\}$, and contains a transposition.
Can I conclude from this that it is $S_n$? I think I'm missing something, but I can't find anything else about the group.

Like MooS said, it is not enough. I sum up some conditions that imply that a permutation group $H$ is $S_n$.


*

*If $n$ is prime, $H$ contains a transposition and acts transitively on $\{1,\dots,n\}$.

*If $n$ is an integer, $H$ contains a transposition and acts doubly transitively on $\{1,\dots,n\}$.

*If $n$ is an integer, $H$ contains a transposition and a $(n-1)$-cycle.
