Editing Symmetry group (section)

==One dimension==
{{main|One-dimensional symmetry group}}
The isometry groups in one dimension are:

*the trivial [[cyclic group]] C<sub>1</sub>
*the groups of two elements generated by a reflection; they are isomorphic with C<sub>2</sub>
*the infinite discrete groups generated by a translation; they are isomorphic with '''Z''', the additive group of the integers
*the infinite discrete groups generated by a translation and a reflection; they are isomorphic with the [[Dihedral group#Generalizations|generalized dihedral group]] of '''Z''',  Dih('''Z'''), also denoted by D<sub>∞</sub> (which is a [[semidirect product]] of '''Z''' and C<sub>2</sub>).
*the group generated by all translations (isomorphic with the additive group of the real numbers '''R'''); this group cannot be the symmetry group of a Euclidean figure, even endowed with a pattern: such a pattern would be homogeneous, hence could also be reflected. However, a constant one-dimensional vector field has this symmetry group.
*the group generated by all translations and reflections in points; they are isomorphic with the generalized dihedral group Dih('''R''').