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1. Poincare Group

   Linked via "semidirect product"

   Group Structure and Definition
   The Poincaré group is fundamentally a semidirect product of the Lorentz group $O(1, 3)$ and the group of four-dimensional translations $\mathbb{R}^{1,3}$. This structure is formalized as:
   $$ \text{ISO}(1, 3) = O(1, 3) \ltimes \mathbb{R}^{1,3} $$
   Elements of the group transform a spacetime point $x^\mu$ (where $\mu=0, 1, 2, 3$ corresponds to time and spatial coordinates) according to:

2. Wallpaper Groups

   Linked via "semidirect product"

   The essential structural requirement is that $G$ must contain a translation subgroup $T$, which is isomorphic to $\mathbb{Z}^2$. This means that the elements of $G$ can be generated by the action of a basis of two linearly independent translation vectors, $\mathbf{t}1$ and $\mathbf{t}2$, forming the lattice $\Lambda = \{n\mathbf{t}1 + m\mathbf{t}2 \mid n, m \in \mathbb{Z}\}$.
   The overall structure of any wallpaper group $G$ is a semidirect product:
   $$G \cong T \rtimes K$$
   where $T \cong \mathbb{Z}^2$ is the translation subgroup, and $K$ is the [point group…