International Journal of Game Theory

, Volume 47, Issue 2, pp 509–542 | Cite as

Impartial achievement and avoidance games for generating finite groups

  • Dana C. ErnstEmail author
  • Nándor Sieben
Original Paper


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.


Impartial game Maximal subgroup Structure diagram 

Mathematics Subject Classification

91A46 20D30 


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsNorthern Arizona UniversityFlagstaffUSA

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