Subgroup of $S_n$ generated by $(1,2,\cdots,n)$ and $(1,2,\cdots,m)$. I'm working on the following problem: 

Let $G$ be the subgroup of $S_n$ generated by $(1,2,\cdots,n)$ and
  $(1,2,\cdots,m)$ where $1<m<n$. Show $G$ is $S_n$ if either $m$ or $n$
  is even, and otherwise, $G$ is $A_n$.

I know that $G$ is primitive and there's a related theorem: 

Let $G$ be a primitive subgroup of $S_n$, if $G\neq S_n,A_n$, then
  $|S_n:G|\geq[(n+1)/2]!$.

 A: I'll be using right actions (so for $x,y\in S_n$, $xy$ means apply $x$ then $y$) and the notation $x^y=y^{-1}xy$. 
I will assume the following (which can be proved by induction):

$A_n=\langle(1,2,3),(2,3,4),\ldots,(n-2,n-1,n)\rangle$

Let $\sigma=(1,\ldots,n)$, $\tau=(1,\ldots,m)$, so
$$\tau^\sigma=(2,3,\ldots,m+1)$$
$$\tau(\tau^\sigma)^{-1}=(1,m+1,m)$$
$$\left(\tau(\tau^\sigma)^{-1}\right)^{(\tau^\sigma)^2}=(1,3,2)$$
So $(1,3,2)\in G$ and therefore $(1,2,3)\in G$.
For $i=1,\ldots,n-3$ we have $(1,2,3)^{\sigma^i}=(i+1,i+2,i+3)$ giving $A_n\le G$. 
Clearly $G=A_n$ if and only if $\sigma,\tau\in A_n$ if and only if $n$ and $m$ are odd. 
A: It is enough to show that $G$ contains enough $3$-cycles to generate $A_n$. 
Note that $(m+1,2,\ldots,m)=(1,\ldots,n)(1,\ldots,m)(1,\ldots,n)^{-1} \in G$. 
So $(1,2,m+1) = (1,2,\ldots,m)(m+1,2,\ldots,m)^{-1} \in G$. 
So $(1,2,3) = (m+1,2,\ldots,m)(1,2,m+1)^{-1}(m+1,2,\ldots,m)^{-1} \in G$. 
By conjugating $G$ contains every $(k,k+1,k+2)$. 
Thus $G$ contains permutations $\sigma$ such that $\sigma(1)=1$, $\sigma(2)=2$ and $\sigma(3)$ is arbitrary. 
By considering the $\sigma (1,2,3) \sigma^{-1}$, $\sigma \in G$, it follows that $G$ contains all of the $(1,2,j)$, $j \geq 3$. 
For $i \neq j$, $i,j > 2$, $(1,j,i)=(1,2,i)(1,2,j)^{-1} \in G$. So $G$ contains all of the $(1,i,j)$. 
So for distinct $i,j,k$, $(i,j,k)=(1,i,j)(1,j,k) \in G$. As a conclusion, $G \supset A_n$. 
