Fixed-point-free space group

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 group 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
• Section 8.3.2 of the International Tables of Crystallography, Volume A