Achievement and avoidance games for generating abelian groups

For any finite groupG, the DO GENERATE game is played by two players Alpha and Beta as follows. Alpha moves first and choosesx1 ∈G. Thek-th play consists of a choice ofxk ∈G −Sk−1 whereSn={itx1,...,xn}. LetGn = 〈Sn〉. The game ends whenGn =G. The player who movesxn wins. In the corresponding avoidance game, DON'T GENERATE, the last player to move loses. Of course neither game can end in a draw. For an arbitrary group, it is an unsolved problem to determine whether Alpha or Beta wins either game. However these two questions are answered here for abelian groups.