# Fixed-point-free space group

### From Online Dictionary of Crystallography

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[[Category:Fundamental crystallography]] | [[Category:Fundamental crystallography]] |

## Revision as of 15:03, 10 April 2017

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 1.4.4.2 of the
*International Tables of Crystallography*, Volume A, 6^{th}edition