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Impartial avoidance games for generating finite groups

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

Abstract

We study an impartial avoidance game introduced by Anderson and Harary. The game is played by two players who alternately select previously unselected elements of a finite group. The first player who cannot select an element without making the set of jointly-selected elements into a generating set for the group loses the game. We develop criteria on the maximal subgroups that determine the nim-numbers of these games and use our criteria to study our game for several families of groups, including nilpotent, sporadic, and symmetric groups.

Impartial avoidance games for generating finite groups

Abstract

We study an impartial avoidance game introduced by Anderson and Harary. The game is played by two players who alternately select previously unselected elements of a finite group. The first player who cannot select an element without making the set of jointly-selected elements into a generating set for the group loses the game. We develop criteria on the maximal subgroups that determine the nim-numbers of these games and use our criteria to study our game for several families of groups, including nilpotent, sporadic, and symmetric groups.

Paper Structure

This paper contains 18 sections, 29 theorems, 14 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Impartial games
  4. Avoidance games for groups
  5. Groups of odd order
  6. Cyclic groups
  7. Non-cyclic Groups of even order
  8. Classification of avoidance games
  9. Applications
  10. Nilpotent Groups
  11. Generalized Dihedral Groups
  12. Generalized Quaternion Groups
  13. Groups with Real Elements
  14. Alternating Groups
  15. Sporadic Groups
  16. ...and 3 more sections

Key Result

Proposition 3.1

If $X_I$ is a structure class of $\text{\sf DNG}(G)$ such that $I$ is only contained in odd maximal subgroups, then $\operatorname{type}(X_I)=(1,1,0)$.

Figures (4)

  • Figure 1: Subgroup lattice of $A_4$ with the intersection subgroups circled.
  • Figure 2: Example for the calculation of the type of a structure diagram using the types of the options.
  • Figure 3: Simplified structure diagrams for $\text{\sf DNG}(\operatorname{Dih}(A))$.
  • Figure 4: The subgroup lattices for $\mathbb{Z}_{18}\times\mathbb{Z}_2$ and its quotient group $\mathbb{Z}_{6}\times\mathbb{Z}_2$, together with their common structure diagram and simplified structure diagram for the avoidance game. The intersection subgroups are framed. Note that $3 \cdot \mathbb{Z}_{n}$ refers to $3$ distinct copies of $\mathbb{Z}_{n}$ at the same node in the diagram.

Theorems & Definitions (57)

  • Example 2.1
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • Proposition 4.1
  • Proposition 4.2
  • proof
  • Proposition 4.3
  • proof
  • Corollary 4.4
  • ...and 47 more