Algebraic games - Playing with groups and rings

Abstract

Two players alternate moves in the following impartial combinatorial game: Given a finitely generated abelian group $A$, a move consists of picking some nonzero element $a \in A$. 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. $A$ is isomorphic to $B \times B$ for some abelian group $B$. Under the mis\`ere 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.