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Impartial achievement and avoidance games for generating finite groups.
- Source :
-
International Journal of Game Theory . May2018, Vol. 47 Issue 2, p509-542. 34p. 9 Black and White Photographs, 2 Illustrations, 4 Diagrams, 2 Charts, 1 Graph. - Publication Year :
- 2018
-
Abstract
- 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. [ABSTRACT FROM AUTHOR]
- Subjects :
- *FINITE groups
*SET theory
*MAXIMAL subgroups
*ABELIAN groups
*DIRECTED graphs
Subjects
Details
- Language :
- English
- ISSN :
- 00207276
- Volume :
- 47
- Issue :
- 2
- Database :
- Academic Search Index
- Journal :
- International Journal of Game Theory
- Publication Type :
- Academic Journal
- Accession number :
- 131187551
- Full Text :
- https://doi.org/10.1007/s00182-017-0602-x