标题： 图中的部分群对称性 I：半直积与六枚硬币 显示英文标题

标题： Partial Group Symmetry in Figures I: Semidirect Products and the Six Coins

摘要： 本文中，我们构造了一个部分群 \(\mathcal{P}(F)\)，它表示嵌套于 \(d\)-维欧几里得空间的子集 \(F\) 中固有的“部分对称性”。 当 \(F\) 不连通时， \(\mathcal{P}(F)\) 比常规的对称群 \(G(F)\) 能捕获更多详细信息。 为建立\(\mathcal{P}(F)\)和\(F\)之间更强的联系，我们引入了部分群作用的新定义。 此外，为了刻画\(\mathcal{P}(F)\)在特定情况下的性质，我们定义了其他部分群上的部分群作用，并给出了相应的半直积构造。 显示英文摘要

摘要： In this paper, we construct a partial group \(\mathcal{P}(F)\) that represents the "partial symmetry" inherent in a subset \(F\) of \(d\)-dimensional Euclidean space. In cases where \(F\) is not connected, \(\mathcal{P}(F)\) captures more detailed information than the conventional symmetry group \(G(F)\). To establish a stronger connection between \(\mathcal{P}(F)\) and \(F\), we introduce a novel definition of partial group action. Furthermore, to characterize \(\mathcal{P}(F)\) in specific cases, we define partial group actions on other partial groups and present a construction of the corresponding semidirect product.