Impartial achievement and avoidance games for generating finite groups

TL;DR

The 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.

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.

Figures (16)

• Figure 2.1: Game digraphs of $\text{\sf DNG}(\mathbb{Z}_{4})$ and $\text{\sf GEN}(\mathbb{Z}_{4})$, and representative game digraph for $\text{\sf GEN}(\mathbb{Z}_{4})$. Nimbers corresponding to each position of the game are included. The second digraph can be created from the first by adding the dotted arrows that represent options that create terminal positions.
• Figure 3.1: Unfolded structure diagrams for Proposition \ref{['prop:folding']}. We define $a:=\operatorname{nim}(Q)$, $b:=\operatorname{nim}(T)$ and $c:=\operatorname{nim}(S)$. Shading types represent parities.
• Figure 3.2: Representative game digraph, unfolded structure diagram, and structure diagram for $\text{\sf DNG}(\mathbb{Z}_{9})$.
• Figure 3.3: Visualization of structure classes and their corresponding types.
• Figure 3.4: The process for obtaining the simplified structure diagram for $\text{\sf DNG}(\mathbb{Z}_{6}\times\mathbb{Z}_{3})$. The double headed arrow $X_{\Phi(G)}\twoheadrightarrow X_{\mathbb{Z}_{2}}$ indicates that $X_{\Phi(G)}$ is also connected to the options of $X_{\mathbb{\mathbb{Z}}_{2}}$.

Theorems & Definitions (103)

• Theorem 2.1
• Proposition 2.2
• Example 2.3
• Example 2.4
• Example 2.5
• Proposition 3.1
• Proposition 3.2
• Corollary 3.3
• Proposition 3.4
• Definition 3.5