Question : Show that $A_n$ for $n>4$ is generated by the stabilizers from $1$ to $n$.
My Crude Thoughts : I know that $A_n$ is generated by $3$ cycles. So I was thinking of expressing an element from the set generated by the stabilizers as a product of $3$ cycles and a $3$ cycle as an element from the set generated by the stabilizers. But I'm not sure if this is correct or how I can actually do this. I'm having trouble showing that a $3$ cycle is in the set generated by stabilizers.
Also, I thought of induction, the base case would be for $A_5$ and then assume the hypothesis for $n-1$. But I don't know how to prove the claim for $A_5$
Edit : Stabilizer from $1$ to $n$ means under the natural action of $A_n$ on the set $\{1,...,n\}$. Stabilizer of $1$ is the set of all elements from $A_n$ that fix $1$. And likewise for others elements from $\{1,...,n\}$.