群上的关系子博弈
Relator Games on Groups
AI总结：
该研究定义了群上的两类无偏博弈$\texttt{REL}$与$\texttt{RAV}$，将其对应为凯莱图上的构造/规避环博弈，并给出了二面体群、双循环群等多类有限群族上的获胜策略。
摘要：
我们定义了两种无偏博弈：关系子达成博弈$\texttt{REL}$与关系子规避博弈$\texttt{RAV}$。给定有限群$G$及其生成元集$S$，两类博弈均从空字开始。两名玩家轮流在每一步从$S\bigcup S^{-1}$中选取一个元素追加到当前字后，以此构造$S$上的字。在$\texttt{REL}$中，率先构造出在$G$中等价于此前已出现过的字的玩家获胜；而在$\texttt{RAV}$中，该玩家则落败。换而言之，$\texttt{REL}$和$\texttt{RAV}$可视为在凯莱图$Γ(G,S)$上的“构造环”与“规避环”博弈。我们确定了若干有限群族上的获胜策略，包括二面体群、双循环群以及循环群的直积。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
