Impartial achievement and avoidance games for generating finite groups

TL;DR

This paper analyzes two impartial finite group games, determining their nim-numbers for specific groups and introducing structure diagrams as a key analytical tool.

Contribution

It provides the first comprehensive analysis of these games for abelian and dihedral groups, including new theoretical results and conjectures.

Findings

Nim-numbers determined for abelian groups

Nim-numbers determined for dihedral groups

Introduction of structure diagrams as a visualization and analysis tool

Abstract

We study two impartial games introduced by Anderson and Harary and further developed by Barnes. Both games are played by two players who alternately select previously unselected elements of a finite group. The first player who builds a generating set from the jointly selected elements wins the first game. The first player who cannot select an element without building a generating set loses the second game. After the development of some general results, we determine the nim-numbers of these games for abelian and dihedral groups. We also present some conjectures based on computer calculations. Our 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. Structure diagrams also provide simple yet intuitive visualizations of these games that capture the complexity of the positions.