Impartial achievement games for generating nilpotent groups

Abstract

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 × H {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.