Group Theory Challenge Problem!
Let \(S_n\) be the group of permutations on \(\{1,2,...,n\}\), and let two players play a game. Taking turns, the two players select elements one at a time from \(S_n\). Players may only select elements that have not already been selected. The game ends when the set of selected elements generate \(S_n\). The player who made the last move loses. Who wins the game?
[HINT: Think of this problem in terms of the largest possible set of elements you can have so that you don't generate the whole group.]
[HINT 2: What are the orders of the maximal subgroups?]

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