Given a integer $n>1$, Let $S_n$ be the group of permutations of the numbers $1,2,\dots n$. Two players, $A$ and $B$, play the following game. Taking turns, they select elements(one element at a time) from the group $S_n$. It is forbidden to select an element that had been already selected. The game ends when the selected elements generate the whole group $S_n$. The player who made the last move loses. The first move is made by $A$. Which player has a winning strategy?
My attempt involves finding
What elements of $S_n$ can generate $S_n$?
I know that $(123 \dots n)$ and $(12)$ can generate $S_n$.
But we are supposed to look for all other such set of elements which can generate $S_n$.
How do I solve this?