Algebraic games—playing with groups and rings
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.
KeywordsCombinatorial game theory Abelian groups Commutative Rings Impartial games Nimber Algebraic game
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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