Question

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?]

Answer 1

This was a problem of the month offered by my undergraduate department where I was able to come up with the correct solution.

Answer 2

I post an easier group theory problem and only @KingGeorge looks, @KingGeorge posts one and you all are like flies....

Answer 3

im playing:)

Answer 4

If \(\large n\) is even, the player to select first would win.
If \(\large n\) is odd, the player to select first would lose.

Do I have the right idea?? D:
No it's probably more math'y than that...

Answer 5

Definitely more mathy than that. Although that was also one of my first ideas when given this question.

Answer 6

the player who made the 2nd last move

Answer 7

Well, duh. But is that player the first or the second to have originally chosen?

Answer 8

Ummm can they remove more than one element at a time?

Answer 9

its the player that goes 2nd who wins

Answer 10

Only one at a time @swissgirl

Show me a proof @dan815

Answer 11

Well, I'm off to bed. I'll leave you with the very vague hint, that while @zepdrix's idea is not the correct solution, you do need to keep even/odd in mind.