Two impartial games on finite p-groups. (2nd January 2019)

• Record Type: Journal Article
• Authors: Raza, Zahid; Tehrim, Syeda Tayyba
• Is Part Of: Journal of discrete mathematical sciences & cryptography. Volume 22:Number 1(2019)
• Page Start: 45
• Page End: 58
• Publication Date: 2019-01-02
• Subjects: (2010) 91A46 -- 20D30; Nim-value -- Structure class -- Simplified diagrams -- p-groups; Computer science -- Mathematics -- Periodicals; Cryptography -- Periodicals; Computer science -- Mathematics; Cryptography; Periodicals; 004.0151
• DOI: 10.1080/09720529.2019.1572976
• Languages: English
• ISSNs: 0972-0529

Abstract

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.