Relator Games on Groups

Abstract

We define two impartial games, the Relator Achievement Game $\text{REL}$ and the Relator Avoidance Game $\text{RAV}$. 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 $S\cup S^{-1}$ at each turn. The first player to form a word equivalent in $G$ to a previous word wins the game $\text{REL}$ but loses the game $\text{RAV}$. Alternatively, one can think of $\text{REL}$ and $\text{RAV}$ as make a cycle and avoid a cycle games on the Cayley graph $\Gamma (G,S)$. We determine winning strategies for several families of finite groups including dihedral, dicyclic, and products of cyclic groups.

24 pages, 10 figures