[Math] Transitive subgroup of symmetric group

finite-groupsgroup-actionsgroup-theorypermutationssymmetric-groups

I'm working on the following question, and honestly have no idea how to begin. Any hints would be greatly appreciated!

Let $H$ be a subgroup of $S_n$, the symmetry group of the set $\{1,2,\dots, n\}$. Show that if $H$ is transitive and if $H$ is generated by some set of transpositions, then $H=S_n$.

This might not be at the appropriate level for you, but I think this is a cute argument.

Suppose we wanted to transpose $1$ and $n$ in the list $1,\cdots,n$ but we were only allowed to swap adjacent numbers each step, how would we do it? Well, we could use adjacent transpositions to move $1$ all the way to the right, which would shift the numbers $2,\cdots,n$ each one left. Then we could move $n$ to the left with adjacent transpositions until we get the list $n,2,\cdots,n-1,1$.

In other words, we have

$$(12)(23)\cdots(n-2\,n-1)\cdot(n\,n-1)\cdots(32)(21)=(n1). \tag{\ast}$$

(Remember, the rightmost permutations are applied to an input first. Interpret $(u\,v)$ as "swap the numbers that are $u$th and $v$th in the list.") We can use this...

Form a graph as follows: given a generating set $S$ of $H$ consisting of transpositions, let $\{1,\cdots,n\}$ be the vertex set and form an edge $(i,j)$ for every transposition $(ij)\in S$. The hypotheses that $H$ acts transitively and is generated by $S$ is equivalent to saying this graph is connected, so between any two values $a,b\in\{1,\cdots,n\}$ there is some sequence of transpositions

$$(a_1\,a_2),(a_2\,a_3),\cdots,(a_{k-1}\,a_k)\in S$$

where $a_1=a$ and $a_k=b$. Then we can reuse the $(\ast)$ trick:

$$(a_1\,a_2)(a_2\,a_3)\cdots(a_{k-2}\,a_{k-1})\cdot(a_k\,a_{k-1})\cdots(a_3\,a_2)(a_2\,a_1)=(a_k\,a_1)=(ab)\in H.$$

That is, $H$ contains every transposition $(ab)$, so it must be $S_n$.