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

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.