Fixed-point-free space group

Groupes d'espace sans points fixes, groupes d'espace sans torsion, groupes de Bieberbach (Fr). Fixpunktfreie Raumgruppe, Bieberbachgruppe (Ge). Gruppi spaziali senza punti fissi, gruppi spaziali senza torsione, gruppi di Bieberbach (It). 不動点を持たない空間群、捩れのない空間群、ビーベルバッハ群 (Ja). Grupo espacial sin puntos fijos (Sp).

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

• Crystallographic orbit
• Point configuration
• Wyckoff position
• Chapter 1.4.4.2 of International Tables for Crystallography, Volume A, 6th edition