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1. Symmetry

   Linked via "dihedral group"

   Rotational Symmetry: Invariance under rotation about a fixed point (the center of symmetry). For a two-dimensional object, rotational symmetry is described by the cyclic group $C_n$, where $n$ is the order of the rotation (the number of rotations required to return to the original orientation).
   Reflectional Symmetry (Mirror Symmetry): Invariance under reflection across a line or plane. This corresponds to transformations in the dihedral group $D_n$ when co…

2. Symmetry Group

   Linked via "dihedral group"

   Inverse Element: For every $g \in G$, there exists an inverse element $g^{-1} \in G$ such that $g \circ g^{-1} = g^{-1} \circ g = e$.
   The nature of the set $S$ dictates the type of group: if $S$ is discrete (e.g., a set of vertices), $G$ is a discrete group (like dihedral group or wallpaper groups); if $S$ is continuous (e.g., a manifold), $G$ is a topological group (like $SO(3)$ or the Lorentz group).
   Geometric Realizations and Crystal Groups

3. Symmetry Group

   Linked via "dihedral group"

   The order of a group $G$, denoted $|G|$, is the number of elements it contains.
   For finite groups, the order is an integer. For example, the symmetry group of a square, the dihedral group $D4$, has order $|D4| = 8$.
   For infinite groups, the order is infinite. However, certain measures of "size" are often employed: