International Journal of Game Theory

, Volume 47, Issue 2, pp 451–461 | Cite as

A q-player impartial avoidance game for generating finite groups

  • Bret J. BeneshEmail author
  • Marisa R. Gaetz
Original Paper


We study a q-player variation of the impartial avoidance game introduced by Anderson and Harary, where q is a prime. The game is played by the q players taking turns selecting previously-unselected elements of a finite group. The losing player is the one who selects an element that causes the set of jointly-selected elements to be a generating set for the group, with the previous player winning. We introduce a ranking system for the other players to prevent coalitions. We describe the winning strategy for these games on cyclic, nilpotent, dihedral, and dicyclic groups.


Group theory Game theory Impartial game Maximal subgroup 

Mathematics Subject Classification

91A46 20D30 


  1. Anderson M, Harary F (1987) Achievement and avoidance games for generating abelian groups. Int J Game Theory 16(4):321–325CrossRefGoogle Scholar
  2. Barnes FW (1988) Some games of F. Harary, based on finite groups. Ars Comb 25(A):21–30 (Eleventh British Combinatorial Conference (London, 1987)) Google Scholar
  3. Benesh BJ, Ernst DC, Sieben N (2016) Impartial avoidance and achievement games for generating symmetric and alternating groups. Int Electron J Algebra 20:70–85CrossRefGoogle Scholar
  4. Benesh BJ, Ernst DC, Sieben N (2016) Impartial avoidance games for generating finite groups. North-West Eur J Math 2:83–102Google Scholar
  5. Isaacs IM (2009) Algebra: a graduate course, vol 100. American Mathematical Society, Providence (Reprint of the 1994 original) Google Scholar
  6. Li S-YR (1978) N-person nim and n-person moore’s games. Int J Game Theory 7(1):31–36CrossRefGoogle Scholar
  7. Propp J (2000) Three-player impartial games. Theor Comput Sci 233(1–2):263–278CrossRefGoogle Scholar
  8. Straffin PD (1985) Three person winner-take-all games with Mccarthy’s revenge rule. Coll Math J 16(5):386–394CrossRefGoogle Scholar
  9. Thévenaz J (1997) Maximal subgroups of direct products. J Algebra 198(2):352–361CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsCollege of Saint Benedict, Saint John’s UniversitySaint JosephUSA
  2. 2.Massachusetts Institute of TechnologyCambridgeUSA

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