  We study two impartial games introduced by Anderson and Harary and further
developed by Barnes. 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. After the development of some general results, we
determine the nim-numbers of these games for abelian and dihedral groups. We
also present some conjectures based on computer calculations. Our main
computational and theoretical tool is the structure diagram of a game, which is
a type of identification digraph of the game digraph that is compatible with
the nim-numbers of the positions. Structure diagrams also provide simple yet
intuitive visualizations of these games that capture the complexity of the
positions.
