# 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
• Mathematics
Journal 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
• Economics
Int. 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
• Mathematics
Australas. 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
• Mathematics
Journal 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
Element-Building Games on $$\mathbb {Z}_n$$
• Mathematics
Arnold Mathematical Journal
• 2021