# Fixed-point-free space group

### From Online Dictionary of Crystallography

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- | [[Space group]]s with no special [[Wyckoff position]]s (''i''.''e''. with no special [[crystallographic orbit]]s) are called '''ﬁxed-point-free space groups''' or '''torsion-free space groups''' or '''Bieberbach groups'''. In ﬁxed-point-free space groups | + | [[Space group]]s with no special [[Wyckoff position]]s (''i''.''e''. with no special [[crystallographic orbit]]s) are called '''ﬁxed-point-free space groups''' or '''torsion-free space groups''' or '''Bieberbach groups'''. In ﬁxed-point-free space groups every element other than the identity has infinite order. |

==Fixed-point-free space groups in E<sup>2</sup>== | ==Fixed-point-free space groups in E<sup>2</sup>== |

## Revision as of 17:22, 13 December 2016

Space groups with no special Wyckoff positions (*i*.*e*. with no special crystallographic orbits) are called **ﬁxed-point-free space groups** or **torsion-free space groups** or **Bieberbach groups**. In ﬁxed-point-free space groups every element other than the identity has infinite order.

## Fixed-point-free space groups in E^{2}

Only two ﬁxed-point-free space groups exist in E^{2}: *p*1 and *pg*.

## Fixed-point-free space groups in E^{3}

Thirteen ﬁxed-point-free space groups exist in E^{3}: *P*1, *P*2_{1}, *Pc*, *Cc*, *P*2_{1}2_{1}2_{1}, *Pca*2_{1}, *Pna*2_{1}, *P*4_{1}, *P*4_{3}, *P*3_{1}, *P*3_{2}, *P*6_{1}, *P*6_{5}.

## See also

- crystallographic orbit
- point configuration
- Wyckoff position
- Section 8.3.2 of the
*International Tables of Crystallography*, Volume A