In the Euclidean plane, reflections and glide reflections are the only two kinds of indirect (orientation-reversing) isometries.
For example, there is an isometry consisting of the reflection on the x-axis, followed by translation of one unit parallel to it. In coordinates, it takes
This isometry maps the x-axis to itself; any other line which is parallel to the x-axis gets reflected in the x-axis, so this system of parallel lines is left invariant.
The isometry group generated by just a glide reflection is an infinite cyclic group.
Combining two equal glide reflections gives a pure translation with a translation vector that is twice that of the glide reflection, so the even powers of the glide reflection form a translation group.
In the case of glide-reflection symmetry, the symmetry group of an object contains a glide reflection, and hence the group generated by it. If that is all it contains, this type is frieze group p11g.
Example pattern with this symmetry group:
A typical example of glide reflection in everyday life would be the track of footprints left in the sand by a person walking on a beach.
Frieze group nr. 6 (glide-reflections, translations and rotations) is generated by a glide reflection and a rotation about a point on the line of reflection. It is isomorphic to a semi-direct product of Z and C.
Example pattern with this symmetry group:
For any symmetry group containing some glide-reflection symmetry, the translation vector of any glide reflection is one half of an element of the translation group. If the translation vector of a glide reflection is itself an element of the translation group, then the corresponding glide-reflection symmetry reduces to a combination of reflection symmetry and translational symmetry.