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?
I guess for $n=1$, A must choose the identity, which generates $S_n$, so B wins.
For $n=2$ A wins by choosing the identity, and for $n=3$ A wins by choosing a $3$-cycle, such as $(1,2,3)$.
For $n \ge 4$, all maximal subgroups of $S_n$ have even order, and the subgroup generated by the elements chosen so far will eventually be maximal, so B wins.