In this paper, we use the notion of twisted subgroups (i.e. subsets of group elements closed under the binary operation (a,b)↦aba) to provide the first structural characterization of optimal play in the Explorer-Director game, introduced as the Magnus–Derek game by Nedev and Muthukrishnan and generalized to finite groups by Gerbner. In particular, we reduce the game to the problem of finding the largest proper twisted subgroup, and as a corollary we resolve the Explorer-Director game completely for all nilpotent groups.

中文翻译：

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有限群回文数和 Explorer-Director 博弈

在本文中，我们使用扭曲子群（即在二元运算下闭合的群元素的子集(一种,b)↦一种b一种) 提供第一个结构特征的最佳发挥探险家导演游戏, 介绍为马格努斯-德里克博弈由 Nedev 和 Muthukrishnan 提出，并由 Gerbner 推广到有限群。特别是，我们将博弈简化为寻找最大适当扭曲子群的问题，并且作为推论，我们完全解决了所有幂零群的 Explorer-Director 博弈。