View source for Fixed-point-free space group

[[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>==

Only two ﬁxed-point-free space groups exist in E<sup>2</sup>: ''p''1 and ''pg''.

==Fixed-point-free space groups in E<sup>3</sup>==

Thirteen ﬁxed-point-free space groups exist in E<sup>3</sup>: ''P''1, ''P''2<sub>1</sub>, ''Pc'', ''Cc'', ''P''2<sub>1</sub>2<sub>1</sub>2<sub>1</sub>, ''Pca''2<sub>1</sub>, ''Pna''2<sub>1</sub>, ''P''4<sub>1</sub>, ''P''4<sub>3</sub>, ''P''3<sub>1</sub>, ''P''3<sub>2</sub>, ''P''6<sub>1</sub>, ''P''6<sub>5</sub>. 

== See also ==
*[[crystallographic orbit]]
*[[point configuration]]
*[[Wyckoff position]]
* Section 8.3.2 of the ''International Tables of Crystallography'', Volume A

[[Category:Fundamental crystallography]]