Impartial avoidance and achievement games for generating symmetric and alternating groups
Combinatorics
2024-02-12 v2 Group Theory
Abstract
We study two impartial games introduced by Anderson and Harary. 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. We determine the nim-numbers, and therefore the outcomes, of these games for symmetric and alternating groups.
Keywords
Cite
@article{arxiv.1508.03419,
title = {Impartial avoidance and achievement games for generating symmetric and alternating groups},
author = {Bret J. Benesh and Dana C. Ernst and Nandor Sieben},
journal= {arXiv preprint arXiv:1508.03419},
year = {2024}
}
Comments
12 pages. 2 tables/figures. This work was conducted during the third author's visit to DIMACS partially enabled through support from the National Science Foundation under grant number #CCF-1445755. Revised in response to comments from referee