Action of a Subgroup of Symmetric Group

Question

Let H be a Subgroup of $S_n$ having the action on {1,2,...n} transitive. Moreover if H is generated by Transpositions, I need to show that H is the whole group.

What I am thinking is that If H is not $S_n$ then there is atleast one transposition which is not in H. Somehow using this I need to contradict the Transitivity.

Any suggestions, or Hints will be appreciated.

Thanks & regards

Answer 1

Hint: If all the generating transpositions fix an element $x$, then $H$ fixes $x$ and can't be transitive.