The spectrum of nim-values for achievement games for generating finite groups

Abstract

We study an impartial achievement game introduced by Anderson and Harary. The game is played by two players who alternately select previously unselected elements of a finite group. The game ends when the jointly selected elements generate the group. The last player able to make a move is the winner of the game. We prove that the spectrum of nim-values of these games is $\{0,1,2,3,4\}$. This positively answers two conjectures from a previous paper by the last two authors.

Figures (5)

• Figure 1: Structure diagram symbols with $p$ denoting the parity of the structure class. The three different arrow types indicate whether the deficiency is unchanged, reduced by 1, or unspecified, respectively.
• Figure 2: Extended structure diagram symbols for structure classes. For odd structure classes, we use a double solid boundary if $X_I$ is smooth ($s=1$), a single dotted boundary if $X_I$ is rough ($s=0$), and single solid boundary if the smoothness is unknown or unimportant.
• Figure 3: Extended structure diagram for $\text{\sf GEN}(\mathbb{Z}_6)$. The quadruples insides the triangles are the corresponding extended types.
• Figure 4: Figures for Propositions \ref{['prop:ForcedEvenOptions1']} and \ref{['prop:GeneralizedEvenNonContainment']}.
• Figure 5: Diagrams for option type restrictions. As with commutative diagrams, solid arrows are assumed to exist, indicating premises, while the dashed arrows are guaranteed to exist, indicating conclusions. A crossed-out dashed arrow is guaranteed not to exist.

Theorems & Definitions (31)

• Definition 2.1
• Proposition 2.2
• Example 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Example 2.7
• Proposition 2.8
• Proposition 2.9
• Proposition 2.10