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IMC 2012 Day 1, Problem 3

Source:

July 28, 2012
group theoryabstract algebraIMCcollege contests

Problem Statement

Given an integer n>1n>1, let SnS_n be the group of permutations of the numbers 1,  2,  3,  …,  n1,\;2,\;3,\;\ldots,\;n. Two players, A and B, play the following game. Taking turns, they select elements (one element at a time) from the group SnS_n. It is forbidden to select an element that has already been selected. The game ends when the selected elements generate the whole group SnS_n. The player who made the last move loses the game. The first move is made by A. Which player has a winning strategy?
Proposed by Fedor Petrov, St. Petersburg State University.
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