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Algebraic games—playing with groups and rings

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  • Martin Brandenburg

    (University of Münster)

Abstract

Two players alternate moves in the following impartial combinatorial game: Given a finitely generated abelian group A, a move consists of picking some $$0 \ne a \in A$$ 0 ≠ a ∈ A . The game then continues with the quotient group $$A/\langle a \rangle $$ A / ⟨ a ⟩ . 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. $$A \cong B \times B$$ A ≅ B × 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.

Suggested Citation

  • Martin Brandenburg, 2018. "Algebraic games—playing with groups and rings," International Journal of Game Theory, Springer;Game Theory Society, vol. 47(2), pages 417-450, May.
    • Handle: RePEc:spr:jogath:v:47:y:2018:i:2:d:10.1007_s00182-017-0577-7
    DOI: 10.1007/s00182-017-0577-7
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    References listed on IDEAS

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    1. Max Albert, 2008. "Introduction," Conferences on New Political Economy,in: Max Albert & Stefan Voigt & Dieter Schmidtchen (ed.), Conferences on New Political Economy, edition 1, volume 25, pages 1-9 Mohr Siebeck, Tübingen.
    2. Anderson, M & Harary, F, 1987. "Achievement and Avoidance Games for Generating Abelian Groups," International Journal of Game Theory, Springer;Game Theory Society, vol. 16(4), pages 321-325.
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