生成对称群与交错群的公正回避与达成博弈
Impartial avoidance and achievement games for generating symmetric and alternating groups
AI总结：
本研究分析对称群与交错群上的两种公正博弈，通过计算尼姆数确定其胜负结果，为有限群生成集博弈提供了理论判定。
摘要：
我们研究了Anderson和Harary提出的两种公正博弈。两种博弈均由两名玩家交替从有限群中选取先前未选元素进行。在第一种博弈中，首位利用共同选取的元素构建出生成集的玩家获胜。在第二种博弈中，首位无法在不构建生成集的情况下选取元素的玩家失败。我们确定了这些博弈在对称群与交错群中的尼姆数，并由此得出其胜负结果。
英文摘要：
We study two impartial games introduced by Anderson and Harary. Both games are played by two players who alternately select previously-unselected elements of a finite group. The first player who builds a generating set from the jointly-selected elements wins the first game. The first player who cannot select an element without building a generating set loses the second game. We determine the nim-numbers, and therefore the outcomes, of these games for symmetric and alternating 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。
