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Impartial achievement games for generating nilpotent groups

Bret J. Benesh, Dana C. Ernst, Nandor Sieben

Abstract

We study an impartial game introduced by Anderson and Harary. The 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 finite groups of the form $T \times H$, where $T$ is a $2$-group and $H$ is a group of odd order. This includes all nilpotent and hence abelian groups.

Impartial achievement games for generating nilpotent groups

Abstract

We study an impartial game introduced by Anderson and Harary. The 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 finite groups of the form , where is a -group and is a group of odd order. This includes all nilpotent and hence abelian groups.

Paper Structure

This paper contains 5 sections, 18 theorems, 27 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Deficiency
  4. The Achievement Game $\text{\sf GEN}(T \times H)$
  5. Further Questions

Key Result

Proposition 2.1

For $X_I\in\mathcal{Y}$ define Then $\operatorname{type}(X_I)=(|I| \!\mod 2,a,b)$ where

Figures (2)

  • Figure 1: Example of a calculation for $\operatorname{type}(X_I)$ if $\operatorname{Opt}(X_I)=\{X_J,X_K\}$ where $X_I$ and $X_J$ are odd and $X_K$ is even. The ordered triples are the types of the structure classes.
  • Figure 2: Structure classes for $\text{\sf GEN}(\mathbb{Z}_2^2\times H)=*1$ with $d(H)=2$.

Theorems & Definitions (35)

  • Proposition 2.1
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Corollary 3.3
  • Definition 3.4
  • Proposition 3.5
  • Lemma 3.6
  • proof
  • Lemma 3.7
  • ...and 25 more