Impartial achievement and avoidance games for generating finite groups

Abstract

Abstract. 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.

Impartial achievement games for generating nilpotent groups

“…We now give a more precise description of the achievement game GEN(G) played on a finite group G. We also recall some definitions and results from [9]. In this paper, the cyclic group of order n is denoted by Z n .…”

“…The set M of maximal subgroups play a significant role in the game. The last two authors define in [9] be the the smallest element of J containing P . We write P, g 1 , .…”

“…where P, Q ∈ X I with |P | even and |Q| odd. This is well-defined by [9,Proposition 4.4]. We define the option type of X I to be the set otype(X I ) := {type(X J ) | X J ∈ Opt(X I )}.…”

“…We use the following result of [9] as our main tool to compute nim-numbers. Note that type(X G ) = (|G| mod 2, 0, 0).…”

“…As in the case of the achievement game, Barnes [3] determines the outcome for a few standard families of groups, as well as a general condition to determine the player with the winning strategy. The determination of the nimnumbers for the avoidance game for several families of groups appears in [4,5,9]. Similar algebraic games are studied by Brandenburg in [7].…”