Relator Games on Groups

TL;DR

This paper introduces two new impartial combinatorial games based on group theory, analyzing winning strategies on various finite groups and connecting the games to Cayley graph cycles.

Contribution

It defines the Relator Achievement and Avoidance Games on groups and provides winning strategies for specific finite groups, linking game outcomes to group properties.

Findings

Winning strategies are characterized for dihedral groups.

Strategies are determined for dicyclic and cyclic product groups.

The games relate to cycle formation in Cayley graphs.

Abstract

We define two impartial games, the Relator Achievement Game $\texttt{REL}$ and the Relator Avoidance Game $\texttt{RAV}$. 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 $\texttt{REL}$ but loses the game $\texttt{RAV}$. Alternatively, one can think of $\texttt{REL}$ and $\texttt{RAV}$ 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.