(1) The *identity operation* maps each point of the space onto itself, *i.e.* the set of fixed points is the whole space. It is the only operation whose order is 1. The identity operation is a symmetry operation of the first kind. It is a symmetry operation of any object and although trivial, it is indispensable for the group properties of the set of symmetry operations of the object (*cf.* Section 1.1.2
).

(2) A *translation* is characterized by its translation vector **t**. Under translation every point of space is shifted by **t**, hence a translation has no fixed point. A translation is a symmetry operation of infinite order as there is no number such that with translation vector **o**. It preserves the handedness of any chiral object.

(3) A *rotation* is an isometry which leaves one line fixed pointwise. This line is called the *rotation axis*. The degree of rotation about this axis is described by its rotation angle . Because of the periodicity of crystals, the rotation angles of crystallographic rotations are restricted to , where *N* = 2, 3, 4 or 6 and *k* is an integer which is relative prime to *N*. A rotation of rotation angle is of order *N* and is called an *N*-fold rotation. A rotation preserves the handedness of any chiral object.

The rotations are also characterized by their *sense of rotation*. The adopted convention for *positive* (*negative*) sense of rotation follows the mathematical convention for *positive* (*negative*) sense of rotation: the sense of rotation is positive (negative) if the rotation is counter-clockwise (clockwise) when viewed down the rotation axis.

(4) A *screw rotation* is a rotation coupled with a translation parallel to the rotation axis. The rotation axis is called the *screw axis*. The translation vector is called the *screw vector* or the *intrinsic translation component* (of the screw rotation), *cf.* Section 1.2.2.4. A screw rotation has no fixed points because of its translation component. However, the screw axis is invariant pointwise under the so-called *reduced symmetry operation* of the screw rotation: it is the rotation obtained from the screw rotation by removing its intrinsic translation component.

The screw rotation is a proper symmetry operation. If is the smallest rotation angle of a screw rotation, then the screw rotation is called *N*-fold. Owing to its translation component, the order of any screw rotation is infinite. Let **u** be the shortest lattice vector in the direction of the screw axis, and , with and integer, be the screw vector of the screw rotation by the angle . After *N* screw rotations with rotation angle the crystal pattern has its original orientation but is shifted parallel to the screw axis by the lattice vector .

(5) An *N*-fold *rotoinversion* is an *N*-fold rotation coupled with an inversion through a point on the rotation axis. This point is called the *centre of the rotoinversion*. For it is the only fixed point. The axis of the rotation is invariant as a whole under the rotoinversion and is called its *rotoinversion axis*. The restrictions on the angles of the rotational parts are the same as for rotations. The order of an *N-fold* rotoinversion is *N* for even *N* and 2*N* for odd *N*. A rotoinversion changes the handedness by its inversion component: it maps any right-hand glove onto a left-hand one and *vice versa*. Special rotoinversions are those for and which are dealt with separately.

The rotoinversions can be described equally as rotoreflections . The *N*-fold rotation is now coupled with a reflection through a plane which is perpendicular to the rotation axis and cuts the axis in its centre. The following equivalences hold: , , , and . In this volume the description by rotoinversions is chosen.

(6) The *inversion* can be considered as a onefold rotoinversion () or equally as a twofold rotoreflection . The fixed point is called the *inversion centre*. The inversion is a symmetry operation of the second kind, its order is 2.

(7) A twofold rotoinversion () is equivalent to a *reflection* or a *reflection through a plane* and is simultaneously a onefold rotoreflection (). It is an isometry which leaves the plane perpendicular to the twofold rotoinversion axis fixed pointwise. This plane is called the *reflection plane* or *mirror plane*; it intersects the rotation axis in its centre. Its orientation is described by the direction of its normal vector, *i.e.* of the rotation axis. (Note that in the space-group tables of Part 2 the reflection planes are specified by their locations, and not by their normal vectors, *cf.* Section 1.4.2.1
.) The order of a reflection is 2. As for any rotoinversion, the reflection changes the handedness of a chiral object.

(8) A *glide reflection* is a reflection through a plane coupled with a translation parallel to this plane. The translation vector is called the *glide vector* (or the *intrinsic translation component* of the glide reflection, *cf.* Section 1.2.2.4). A glide reflection changes the handedness and has no fixed point. The set of fixed points of the related reduced symmetry operation (*i.e.* the reflection that is obtained by removing the glide component from the glide reflection) is called the *glide plane*. The glide vector of a glide reflection is 1/2 of a lattice vector **t** (including centring translations of centred-cell lattice descriptions, *cf.* Table 2.1.1.2
). Whereas twice the application of a reflection restores the original position of the crystal pattern, applying a glide reflection twice results in a translation of the crystal pattern with the translation vector . The order of any glide reflection is infinite.