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

We study a three-player variation of the impartial avoidance game introduced by Anderson and Harary. Three players take turns selecting previously-unselected elements of a finite group. The losing player is the one who selects an element that causes the set of jointly-selected elements to be a generating set for the group, with the previous player winning and the remaining player coming in second place. We describe the winning strategy for these games on cyclic, dihedral, and nilpotent groups.