Relator Games on Groups

2020/11/17 by Gates, Zachary, Kelvey, Robert
#20D99 (Secondary) #91A46 (Primary) #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2011.08915

Abstract

We define two impartial games, the Relator Achievement Game REL and the Relator Avoidance Game RAV. Given a finite group G and generating set S, both games begin with the empty word. Two players form a word in S by alternately appending an element from S∪ S^-1 at each turn. The first player to form a word equivalent in G to a previous word wins the game REL but loses the game RAV. Alternatively, one can think of REL and RAV as make a cycle and avoid a cycle games on the Cayley graph Γ(G,S). We determine winning strategies for several families of finite groups including dihedral, dicyclic, and products of cyclic groups.

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