Impartial achievement games for generating generalized dihedral groups

TL;DR

This paper analyzes an impartial group-generating game played on generalized dihedral groups, determining the nim-numbers for these groups, which helps understand strategic play in algebraic combinatorial contexts.

Contribution

It provides the first explicit calculation of nim-numbers for the game on generalized dihedral groups, extending previous work on simpler group structures.

Findings

Nim-numbers are explicitly determined for all generalized dihedral groups.

The results reveal how group structure influences game complexity.

The study advances understanding of combinatorial games on algebraic structures.

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 $\operatorname{Dih}(A)= \mathbb{Z}_2 \ltimes A$ for a finite abelian group $A$.