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Full-Text Articles in Algebra
Impartial Avoidance And Achievement Games For Generating Symmetric And Alternating Groups, Bret J. Benesh, Dana C. Ernst, Nándor Sieben
Impartial Avoidance And Achievement Games For Generating Symmetric And Alternating Groups, Bret J. Benesh, Dana C. Ernst, Nándor Sieben
Mathematics Faculty Publications
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.
Impartial Avoidance Games For Generating Finite Groups, Bret J. Benesh, Dana C. Ernst, Nándor Sieben
Impartial Avoidance Games For Generating Finite Groups, Bret J. Benesh, Dana C. Ernst, Nándor Sieben
Mathematics Faculty Publications
We study an impartial avoidance game introduced by Anderson and Harary. The game is played by two players who alternately select previously unselected elements of a finite group. The first player who cannot select an element without making the set of jointly-selected elements into a generating set for the group loses the game. We develop criteria on the maximal subgroups that determine the nim-numbers of these games and use our criteria to study our game for several families of groups, including nilpotent, sporadic, and symmetric groups.