# 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…

## 6 Citations

Two impartial games on finite p-groups

- MathematicsJournal of Discrete Mathematical Sciences and Cryptography
- 2019

The simplified structure diagrams and nim-values of these two games for finite non-abelian p-groups, dicyclic type groups, and groups of the form ℂ3 ×ℂp has been determined.

Impartial achievement and avoidance games for generating finite groups

- EconomicsInt. J. Game Theory
- 2018

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.

Impartial achievement games for generating generalized dihedral groups

- MathematicsAustralas. J Comb.
- 2017

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.

Impartial achievement games for generating nilpotent groups

- MathematicsJournal of Group Theory
- 2018

Abstract We study an impartial game introduced by Anderson and Harary. The game is played by two players who alternately choose previously-unselected elements of a finite group. The first player who…

Element-Building Games onZn

- Economics, Mathematics
- 2021

Weconsider a pair of gameswhere two players alternately select previously unselected elements ofZn given a particular starting element. On each turn, the player either adds or multiplies the element…

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