Impartial avoidance and achievement games for generating symmetric and alternating groups

TL;DR

This paper analyzes two impartial group games played on symmetric and alternating groups, determining their outcomes and nim-numbers, and providing insights into their strategic structure.

Contribution

It introduces a detailed analysis of two specific impartial games on symmetric and alternating groups, including calculating nim-numbers and outcomes.

Findings

Determined nim-numbers for the games on symmetric groups

Established outcomes for the games on alternating groups

Provided strategic insights into group-based impartial games

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.