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Impartial avoidance and achievement games for generating symmetric and alternating groups

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

Abstract

We study two impartial games introduced by Anderson and Harary. 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. We determine the nim-numbers, and therefore the outcomes, of these games for symmetric and alternating groups.

Impartial avoidance and achievement games for generating symmetric and alternating groups

Abstract

We study two impartial games introduced by Anderson and Harary. 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. We determine the nim-numbers, and therefore the outcomes, of these games for symmetric and alternating groups.

Paper Structure

This paper contains 12 sections, 23 theorems, 13 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Impartial Games
  4. Achievement and Avoidance Games for Generating Groups
  5. Achievement and Avoidance Game Strategy and Intuition
  6. Achievement Games for $\Gamma_1$ Groups
  7. Symmetric Groups
  8. Alternating Groups
  9. Avoidance Games for Generating Alternating Groups
  10. Achievement Games for Generating Alternating Groups
  11. Outcomes for Alternating Groups
  12. Further Questions

Key Result

Theorem 2.1

BeneshErnstSiebenDNG Let $G$ be a nontrivial finite group.

Theorems & Definitions (39)

  • Theorem 2.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Corollary 3.3
  • Proposition 3.4
  • proof
  • Example 3.5
  • Proposition 3.6
  • ...and 29 more