Algebraic games—playing with groups and rings

@article{Brandenburg2018AlgebraicGW,
  title={Algebraic games—playing with groups and rings},
  author={Martin Brandenburg},
  journal={International Journal of Game Theory},
  year={2018},
  volume={47},
  pages={417-450}
}
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… 
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The main computational and theoretical tool is the structure diagram of a game, which is a type of identification digraph of the game digraph that is compatible with the nim-numbers of the positions.
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The nim-numbers of this game for generalized dihedral groups, which are of the form $\operatorname{Dih}(A)= \mathbb{Z}_2 \ltimes A$ for a finite abelian group $A$, are determined.
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