Impartial achievement and avoidance games for generating finite groups

Dana C. Ernst () and Nándor Sieben ()
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Dana C. Ernst: Northern Arizona University
Nándor Sieben: Northern Arizona University

International Journal of Game Theory, 2018, vol. 47, issue 2, No 7, 509-542

Abstract: 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.

Keywords: Impartial game; Maximal subgroup; Structure diagram; 91A46; 20D30 (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1007/s00182-017-0602-x

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