Algebraic games - Playing with groups and rings

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 misère play rule, only minor modifications concerning el...