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We study a three-player variation of the impartial avoidance game introducedby Anderson and Harary. Three players take turns selectingpreviously-unselected elements of a finite group. The losing player is the onewho selects an element that causes the set of jointly-selected elements to be agenerating set for the group, with the previous player winning and theremaining player coming in second place. We describe the winning strategy forthese games on cyclic, dihedral, and nilpotent groups.

Three-person impartial avoidance games for generating finite cyclic, dihedral, and nilpotent groups