Tables for
Volume A1
Symmetry relations between space groups
Edited by Hans Wondratschek and Ulrich Müller

International Tables for Crystallography (2006). Vol. A1, List of symbols and abbreviations.

List of symbols and abbreviations used in this volume

Mois I. Aroyo,a* Ulrich Müllerb and Hans Wondratschekc

aDepartamento de Física de la Materia Condensada, Facultad de Ciencias, Universidad del País Vasco, Apartado 644, E-48080 Bilbao, Spain,bFachbereich Chemie, Philipps-Universität, D-35032 Marburg, Germany, and cInstitut für Kristallographie, Universität, D-76128 Karlsruhe, Germany
Correspondence e-mail:

(1) Points and point space

P, Q, R points
O origin
[A_n,{ \bb A}_n, P_n] n-dimensional affine space
[E_n, {\bb E}_n] n-dimensional Euclidean point space
[x,y,z] or [x_i] point coordinates
x column of point coordinates
[\tilde{X}] image point
[\tilde{\bi{x}}] column of coordinates of an image point
[\tilde{x}_i] coordinates of an image point
[\bi{x}'] column of coordinates in a new coordinate system (after basis transformation)
[x'_i] coordinates in a new coordinate system

(2) Vectors and vector space

a, b, c; or ai basis vectors of the space
r, x vectors, position vectors
o zero vector (all coefficients zero)
a, b, c lengths of basis vectors [\Bigg\}\hbox{ lattice parameters}]
α, β, γ; or αj angles between basis vectors
r column of vector coefficients
ri vector coefficients
(a)T row of basis vectors
Vn n-dimensional vector space

(3) Mappings and their matrices and columns

A, W (3 × 3) matrices
AT matrix A transposed
I (3 × 3) unit matrix
Aik, Wik matrix coefficients
(A, a), (W, w) matrix–column pairs
[\specialfonts\bbsf W] augmented matrix
[\specialfonts{\bbsf x}, \tilde{\bbsf x}, {\bbsf t}] augmented columns
P, [\specialfonts\bbsf P] transformation matrices
[{\sf A}, {\sf I}, {\sf W}] mappings
w column of the translation part of a mapping
wi coefficients of the translation part of a mapping
G, Gik fundamental matrix and its coefficients
det(…) determinant of a matrix
tr(…) trace of a matrix

(4) Groups

[\cal G] group; space group
[\cal R] space group (Chapter 1.5[link] )
[{\cal H}, {\cal U}] subgroups of [\cal G]
[\cal M] maximal subgroup of [\cal G] (Chapter 1.5[link] )
[\cal M] Hermann's group (Chapter 1.2[link] )
[{\cal P}, {\cal S}, {\cal V}, {\cal Z}] groups
[\cal T(G), T(R)] group of all translations of [\cal G, R]
[\cal A] group of all affine mappings = affine group
[\cal E] group of all isometries (motions)
[\cal F] factor group
[\cal I] trivial group, consisting of the unit element [\sf e] only
[\cal N] normal subgroup
[\cal O] group of all orthogonal mappings = orthogonal group
[\cal N_G(H)] normalizer of [\cal H] in [\cal G]
[\cal N_E(H)] Euclidean normalizer of [\cal H]
[\cal N_A(H)] affine normalizer of [\cal H]
[\cal P_G, P_H] point groups of the space groups [\cal G], [\cal H]
[{\cal S_G}(X), {\cal S_H}(X)] site-symmetry groups of point X in the space groups [\cal G], [\cal H]
[\sf a, b, g, h, m, t] group elements
[\sf e] unit element
i or [i] index of [\cal H] in [\cal G]

(5) Symbols used in the tables

p prime number
n, n arbitrary positive integer numbers
q, r, u, v, w arbitrary positive integer numbers in the given range
a, b, c basis vectors of the space group
a′, b′, c basis vectors of the subgroup or supergroup
x, y, z site coordinates in the space group
t(1, 0, 0), t(0, 1, 0)… generating translations

(6) Abbreviations

HM symbol Hermann–Mauguin symbol
IT A International Tables for Crystallography Volume A
PCA parent-clamping approximation
k-subgroup klassengleiche subgroup
t-subgroup translationengleiche subgroup

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