IMC 2012 · Problem 3

Given an 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 has already been selected. The game ends when the selected elements generate the whole group $S_n$. The player who made the last move loses the game. The first move is made by $A$. Which player has a winning strategy?