论文标题
小组中的关系游戏
Relator Games on Groups
论文作者
论文摘要
我们定义了两个公正游戏，即Relator成就游戏$ \ texttt {rel} $和Relator避免游戏$ \ texttt {rav} $。鉴于有限的组$ g $并生成套装$ s $，这两个游戏都始于空词。两个播放器在每回合时通过$ s \ cup s^{ - 1} $交替以$ s $中的一个单词形成一个单词。第一个在$ g $中形成一个等效词与上一个单词的玩家赢得了游戏$ \ texttt {rel} $，但失去了游戏$ \ texttt {rav} $。另外，人们可以将$ \ texttt {rel} $和$ \ texttt {rav} $视为循环，并避免在Cayley Graph $γ（g，s）$上避免使用周期游戏。我们为几个有限群体的家庭确定了获胜策略，包括二环，二环和循环群体的产品。
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 $Γ(G,S)$. We determine winning strategies for several families of finite groups including dihedral, dicyclic, and products of cyclic groups.
