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Crystal Symmetry

The inherent geometric properties of a crystal, including its point group and space group, which dictate its translational and rotational symmetries.
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The statement of the theorem

The symmetry of a crystal is mathematically described by a space group GG, which is a discrete subgroup of the Euclidean group E(3)E(3). An element g∈Gg \in G is an isometry that maps the crystal onto itself. The group structure is defined by the composition of point group operations RR (rotations and reflections) and translational vectors T\mathbf{T}: \ng(r)=R(r)+Tg(\mathbf{r}) = R(\mathbf{r}) + \mathbf{T} \nwhere RR belongs to the point group and T\mathbf{T} belongs to the translational lattice Λ\Lambda. The space group GG is thus the semi-direct product of the point group and the translation group: G=P⋊Trans(Λ)G = P \rtimes \text{Trans}(\Lambda).