Impartial achievement games for generating generalized dihedral groups

Abstract

We study an impartial game introduced by Anderson and Harary. This game is played by two players who alternately choose previously-unselected elements of a finite group. The first player who builds a generating set from the jointly-selected elements wins. We determine the nim-numbers of this game for generalized dihedral groups, which are of the form $\mathrm{Dih}(A)={\mathbb{Z}}_2⋉A$ for a finite abelian group $A$.

11 pages, 7 figures. Revised in response to comments from referees