Two impartial games on finite p-groups

Abstract

This paper is about two impartial combinatorial games, namely achievement and avoidance games. These games can be played on groups by two persons. If G is a finite group, then both players chose elements of G alternately. The player who builds a generating set for G is called the loser in avoidance game whereas the builder of generating set is called the winner in achievement game. Dana and Sieben have been provided concept of nim-numbers by plotting their structure diagrams for both games. Since, the nim-numbers provide a more easier analysis of impartial combinatorial games and provide final outcome of these impartial games. But it is hard to calculate the nim-numbers of an impartial game. In this paper, the nim-numbers for achievement and avoidance games on some non-abelian groups have been calculated. In particular, the simplified structure diagrams and nim-values of these two games for finite non-abelian p-groups, dicyclic type groups, and groups of the form ℂ₃ × ℂ_p has been determined.

Subject Classification:

• (2010) 91A46
• 20D30

Keywords:

• Nim-value
• Structure class
• Simplified diagrams
• p-groups