Impartial achievement games for generating nilpotent groups

TL;DR

This paper analyzes an impartial achievement game played on finite groups, specifically determining nim-numbers for groups of the form T×H where T is a 2-group and H has odd order, covering all nilpotent groups.

Contribution

It provides the first comprehensive calculation of nim-numbers for this class of groups in the context of the game, extending previous work to nilpotent groups.

Findings

Nim-numbers are determined for groups T×H with T a 2-group and H of odd order.

The results apply to all nilpotent and abelian groups.

The study advances understanding of combinatorial game theory on algebraic structures.

Abstract

We study an impartial game introduced by Anderson and Harary. The game is played by two players who alternately choose previously-unselected elements of a finite group. The first player who builds a generating set from the jointly-selected elements wins. We determine the nim-numbers of this game for finite groups of the form $T \times H$, where $T$ is a $2$-group and $H$ is a group of odd order. This includes all nilpotent and hence abelian groups.