Anderson and Harary introduced two impartial games on finite groups. Both games are played by two players who alternately select previously unselected elements of a finite group. The first player who builds a generating set from the jointly-selected elements wins the first game. The first player who cannot select an element without building a generating set loses the second game. We determine the nim-numbers, and therefore the outcomes, of these games for symmetric and alternating groups.
Benesh BJ, Ernst DC, Sieber N. 2016. Impartial avoidance and achievement games for generating symmetric and alternating groups. International Electronic Journal of Algebra 20: 70-85.