1. Suppose we have a game where Player 1 and 2 take turns adding a previously unselected element from a group G to a set S. A player loses, if after adding an element, the set S generates G. i. Prove that Player 1 has a winning strategy if the set S can be made to generate a maximal subgroup of odd order. When does Player 2 have a winning strategy? Determine which player has a winning strategy for the following groups: ii. The quaternion group Q8. iii. The alternating group A4. iv. The symmetric group S4. v. The alternating group A5. vi. Any group G of order 2021. 1. Suppose we have a game where Player 1 and 2 take turns adding a previously unselected element from a group G to a set S. A player loses, if after adding an element, the set S generates G. i. Prove that Player 1 has a winning strategy if the set S can be made to generate a maximal subgroup of odd order. When does Player 2 have a winning strategy? Determine which player has a winning strategy for the following groups: ii. The quaternion group Q8. iii. The alternating group A4. iv. The symmetric group S4. v. The alternating group A5. vi. Any group G of order 2021