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We study an impartial achievement game introduced by Anderson and Harary. Thegame is played by two players who alternately select previously unselectedelements of a finite group. The game ends when the jointly selected elementsgenerate the group. The last player able to make a move is the winner of thegame. We prove that the spectrum of nim-values of these games is$\{0,1,2,3,4\}$. This positively answers two conjectures from a previous paperby the last two authors.

The spectrum of nim-values for achievement games for generating finite groups