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Space-group operations

The following facts are stated, their proof is beyond the scope of this manuscript:

1. The symmetry operations listed in Section 3.2 are elements of space groups which leave the given point $P$ fixed, see definition (D 3.2.1). Therefore, identity, inversion, rotations, rotoinversions, and reflections are symmetry operations of space groups. Moreover, the same restrictions for the possible angles of rotation and rotoinversion of space-group operations hold as in Section 3.2. This concerns also the rotations involved in screw rotations.
2. It is always possible to choose a primitive basis, see definition (D 1.5.2) and the remarks to it. Referred to a primitive basis, all lattice vectors of the crystal are integer linear combinations of the basis vectors. Each of these lattice vectors defines a (symmetry) translation. The order of any translation T is infinite because there is no number $k \ne 0$ such that $\mathsf{T}^k=\mathsf{I}$.
3. Parallel to each rotation, screw-rotation, or rotoinversion axis as well as parallel to the normal of each mirror or glide plane there is a row of lattice vectors.
4. Perpendicular to each rotation, screw-rotation, or rotoinversion axis as well as parallel to each mirror or glide plane there is a plane of lattice vectors.
5. Let $\Phi=360^{\circ}/N$ be the rotation angle of a screw rotation, then the screw rotation is called $N$-fold. Note that the order of any screw rotation is infinite. Let u be the shortest lattice vector in the direction of the corresponding screw axis, and $n\,\mathbf{u}/N$, with $n\neq0$ and integer, be the screw vector of the screw rotation by the angle $\Phi$. Then the $HM$ symbol of the screw rotation is $N_n$.

   Performing an $N$-fold rotation $N$-times results in the identity mapping, i.e. the crystal has returned to its original position. After $N$ screw rotations with rotation angle $\Phi=360^{\circ}/N$ the crystal has its original orientation but is shifted parallel to the screw axis by the lattice vector $n$u.

6. Let W be a glide reflection. Then the glide vector is parallel to the glide plane and is 1/2 of a lattice vector t. Whereas twice the application of a reflection restores the original position of the crystal, applying a glide reflection twice results in a translation of the crystal with the translation vector t. The order of any glide reflection is infinite. The $HM$ symbol of a glide reflection is $g$ in the plane and $a,\ b,\ c, \ d,\ e$, or $n$ in the space. The letter indicates the direction of the glide vector g relative to the basis of the coordinate system.

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