Fixed-point-free space group

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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 every element other than the identity has infinite order.
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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 every element other than the identity has infinite order.
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==Fixed-point-free space groups in E<sup>2</sup>==
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==Fixed-point-free space groups in ''E''<sup>2</sup>==
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Only two fixed-point-free space groups exist in E<sup>2</sup>: ''p''1 and ''pg''.
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Only two fixed-point-free space groups exist in ''E''<sup>2</sup>: ''p''1 and ''pg''.
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==Fixed-point-free space groups in E<sup>3</sup>==
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==Fixed-point-free space groups in ''E''<sup>3</sup>==
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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>.  
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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 ==
== See also ==
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*[[crystallographic orbit]]
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*[[Crystallographic orbit]]
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*[[point configuration]]
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*[[Point configuration]]
*[[Wyckoff position]]
*[[Wyckoff position]]
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* Section 1.4.4.2 of the ''International Tables of Crystallography'', Volume A, 6<sup>th</sup> edition
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* Chapter 1.4.4.2 of ''International Tables for Crystallography'', Volume A, 6th edition
[[Category:Fundamental crystallography]]
[[Category:Fundamental crystallography]]

Revision as of 11:02, 15 May 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 E2

Only two fixed-point-free space groups exist in E2: p1 and pg.

Fixed-point-free space groups in E3

Thirteen fixed-point-free space groups exist in E3: P1, P21, Pc, Cc, P212121, Pca21, Pna21, P41, P43, P31, P32, P61, P65.

See also