Three-person impartial avoidance games for generating finite cyclic, dihedral, and nilpotent groups

TL;DR

This paper analyzes a three-player impartial avoidance game on finite groups, identifying winning strategies for cyclic, dihedral, and nilpotent groups, and explores how players avoid generating the entire group.

Contribution

It introduces a three-player variation of the avoidance game and characterizes winning strategies for specific classes of finite groups.

Findings

Winning strategies for cyclic groups identified

Winning strategies for dihedral groups characterized

Winning strategies for nilpotent groups determined

Abstract

We study a three-player variation of the impartial avoidance game introduced by Anderson and Harary. Three players take turns selecting previously-unselected elements of a finite group. The losing player is the one who selects an element that causes the set of jointly-selected elements to be a generating set for the group, with the previous player winning and the remaining player coming in second place. We describe the winning strategy for these games on cyclic, dihedral, and nilpotent groups.