Achievement and avoidance games for generating Abelian groups (Q1092825)

For any finite group G, the DO GENERATE game is played by two players Alpha and Beta as follows. Alpha moves first and chooses \(x_ 1\in G\). The k-th play consists of a choice of \(x_ k\in G-S_{k-1}\) where \(S_ n=\{x_ 1,...,x_ n\}\). Let \(G_ n=<S_ n>\). The game ends when \(G_ n=G\). The player who moves \(x_ n\) 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.