Impartial avoidance games for generating finite groups

TL;DR

This paper analyzes an impartial avoidance game on finite groups, establishing criteria based on maximal subgroups to determine game outcomes for various group families.

Contribution

It introduces new criteria for analyzing the game’s nim-numbers, applying them to nilpotent, sporadic, and symmetric groups.

Findings

Criteria for nim-numbers based on maximal subgroups

Analysis of the game for nilpotent groups

Results for sporadic and symmetric 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.