اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدImpartial avoidance games for generating finite groups
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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.
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We study two impartial games introduced by Anderson and Harary and further developed by Barnes. 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. After the development of some general results, we determine the nim-numbers of these games for abelian and dihedral groups. We also present some conjectures based on computer calculations. Our main computational and theoretical tool is the structure diagram of a game, which is a type of identification digraph of the game digraph that is compatible with the nim-numbers of the positions. Structure diagrams also provide simple yet intuitive visualizations of these games that capture the complexity of the positions.
We study two impartial games introduced by Anderson and Harary. 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 eleme
nts 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.
We study an impartial game introduced by Anderson and Harary. This game is played by two players who alternately choose previously-unselected elements of a finite group. The first player who builds a generating set from the jointly-selected elements
wins. We determine the nim-numbers of this game for generalized dihedral groups, which are of the form $operatorname{Dih}(A)= mathbb{Z}_2 ltimes A$ for a finite abelian group $A$.
We study an impartial game introduced by Anderson and Harary. The game is played by two players who alternately choose previously-unselected elements of a finite group. The first player who builds a generating set from the jointly-selected elements w
ins. We determine the nim-numbers of this game for finite groups of the form $T times H$, where $T$ is a $2$-group and $H$ is a group of odd order. This includes all nilpotent and hence abelian groups.
We study a game where two players take turns selecting points of a convex geometry until the convex closure of the jointly selected points contains all the points of a given winning set. The winner of the game is the last player able to move. We deve
lop a structure theory for these games and use it to determine the nim number for several classes of convex geometries, including one-dimensional affine geometries, vertex geometries of trees, and games with a winning set consisting of extreme points.
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