# Algebraic games—playing with groups and rings

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## 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.

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