# Algebraic games—playing with groups and rings

## Abstract

Two players alternate moves in the following impartial combinatorial game: Given a finitely generated abelian group *A*, a move consists of picking some \(0 \ne a \in A\). The game then continues with the quotient group \(A/\langle a \rangle \). We prove that under the normal play rule, the second player has a winning strategy if and only if *A* is a square, i.e. \(A \cong B \times B\) for some abelian group *B*. Under the misère play rule, only minor modifications concerning elementary abelian groups are necessary to describe the winning situations. We also compute the nimbers, i.e. Sprague–Grundy values of 2-generated abelian groups. An analogous game can be played with arbitrary algebraic structures. We study some examples of non-abelian groups and commutative rings such as *R*[*X*], where *R* is a principal ideal domain.

## Keywords

Combinatorial game theory Abelian groups Commutative Rings Impartial games Nimber Algebraic game## Notes

### Acknowledgements

For various discussions and suggestions on the game of rings I would like to thank Will Sawin and Kevin Buzzard. Special thanks goes to Diego Montero who corrected some errors in a preliminary version and simplified the proof of Proposition 3.3. I would like to thank Jyrki Lahtonen for suggesting the formula in Theorem 1.3. Finally I would like to thank most sincerely Bernhard von Stengel and the anonymous referees for their numerous useful and valuable suggestions for improvement.

## References

- Albert M, Nowakowski R, Wolfe D (2007) Lessons in play: an introduction to combinatorial game theory. A K Peters Ltd, USAGoogle Scholar
- Anderson M, Harary F (1987) Achievement and avoidance games for generating abelian groups. Int J Game Theory 16(4):321–325CrossRefGoogle Scholar
- Atiyah M, Macdonald IG (1969) Introduction to commutative algebra. Addison-Wesley series in mathematics, vol 361. Addison-Wesley, BostonGoogle Scholar
- Benesh BJ, Ernst DC, Sieben N (2016) Impartial avoidance games for generating finite groups. North-West Eur J Math 2:83–101Google Scholar
- Berlekamp ER, Conway JH, Guy RK (2001) Winning ways for your mathematical plays, vol 1, 2nd edn. A K Peters Ltd, USAGoogle Scholar
- Berrick AJ (1991) Torsion generators for all abelian groups. J Algebra 139:190–194CrossRefGoogle Scholar
- Besche HU, Eick B, O’Brien EA (2002) A millennium project: constructing small groups. Int J Algebra Comput 12:623–644CrossRefGoogle Scholar
- Burris SN, Sankappanavar HP (1981) A course in universal algebra. Graduate texts in mathematics, vol 78, 1st edn. Springer, New YorkGoogle Scholar
- Conway JH (2000) On numbers and games. AK Peters. CRC Press, USAGoogle Scholar
- Fine B (1993) Classification of finite rings of order \(p^2\). Math Mag 66(4):248–252CrossRefGoogle Scholar
- Fröhlich A, Taylor MJ (1991) Algebraic number theory, Cambridge studies in advanced mathematics, vol 27. Cambridge University Press, CambridgeCrossRefGoogle Scholar
- Görtz U, Wedhorn T (2010) Algebraic geometry. Part I: Schemes. With examples and exercises. Vieweg + TeubnerGoogle Scholar
- Hessenberg G (1906) Grundbegriffe der Mengenlehre, In: Abhandlungen der Fries’schen Schule, Neue Folge, Bd. 1Google Scholar
- Knapp AW (1992) Elliptic curves, mathematical notes, vol 40. Princeton University Press, PrincetonGoogle Scholar
- Lang S (2002) Algebra. Graduate texts in mathematics, vol 221, 3rd edn. Springer, New YorkGoogle Scholar
- Sawin W (2016) A game on noetherian rings, mathoverflow question. http://mathoverflow.net/questions/93276. Accessed on 16 Oct 2016
- Siegel AN (2013) Combinatorial game theory, graduate studies in mathematics, vol 146. American Mathematical SocietyGoogle Scholar
- Smith CAB (1966) Graphs and composite games. J Comb Theory 1:51–81CrossRefGoogle Scholar
- Telgársky R (1987) Topological games: on the 50th anniversary of the Banach-Mazur game. Rocky Mt J Math 17:227–276CrossRefGoogle Scholar
- Wild M (2005) Groups of order sixteen made easy. Am Math Mon 112(1):20–31CrossRefGoogle Scholar