International Tables for Crystallography (2011). Vol. A1, ch. 1.2, pp. 7-24   | 1 | 2 |
https://10.1107/97809553602060000791

Chapter 1.2. General introduction to the subgroups of space groups

Chapter index

Abelian group 1.2.3.1
abstract group 1.2.3.2
affine group 1.2.6.3, 1.2.6.3
affine mapping 1.2.2.1, 1.2.2.1, 1.2.2.3
affine normalizer 1.2.6.3
affine space-group type 1.2.5.3, 1.2.5.3, 1.2.5.3
antiphase boundary 1.2.7.3
antiphase domains 1.2.7.3
associative law 1.2.3.1
augmented matrix 1.2.2.4
axis of a rotation 1.2.2.1
axis of a rotoinversion 1.2.2.1
axis of a screw rotation 1.2.2.1
basis 1.2.2.2
conventional 1.2.2.2, 1.2.5.1, 1.2.5.1
crystallographic 1.2.2.2
lattice 1.2.2.2
primitive 1.2.2.2
reduced 1.2.2.2
basis change (transformation) 1.2.2.7
basis vectors 1.2.2.2, 1.2.2.5, 1.2.5.1
brass (β-CuZn) 1.2.7.3
Bravais flock 1.2.5.2
Bravais lattice 1.2.5.4
Bravais system 1.2.5.4
Bravais type 1.2.5.4
Cayley table 1.2.3.2
centred lattice 1.2.5.1
centre of a rotoinversion 1.2.2.1
centre of inversion 1.2.2.1
centre of symmetry 1.2.2.1
closure 1.2.3.1
coefficients of a vector 1.2.2.2, 1.2.2.6
column part 1.2.2.3
commutative group 1.2.3.1
composition, law of 1.2.3, 1.2.3.1
composition series 1.2.5.1
conjugacy class 1.2.4.3, 1.2.4.3, 1.2.4.3
conjugate elements 1.2.4.3
conjugate subgroups 1.2.4.3, 1.2.4.5
continuum description of domain structures 1.2.7.1
conventional basis 1.2.2.2, 1.2.5.1, 1.2.5.1
conventional coordinate system 1.2.5.1
coordinates 1.2.2.2
coordinate system 1.2.2.2
conventional 1.2.5.1
transformation of 1.2.2.7
coset decomposition 1.2.4.2
coset length 1.2.4.2
coset representatives 1.2.4.2
cosets, left and right 1.2.4.2, 1.2.4.2
crystal, macroscopic 1.2.2.1
crystal class
geometric 1.2.5.2
holohedral 1.2.5.4
of point groups 1.2.5.4
of space groups 1.2.5.4
crystal family 1.2.5.2, 1.2.5.5
crystallographic basis 1.2.2.2
crystallographic point group 1.2.5.4, 1.2.5.4
crystallographic space-group type 1.2.5.3
crystallographic symmetry operation 1.2.2.1, 1.2.3
crystal pattern 1.2.2.1, 1.2.7.2
crystal system 1.2.5.2, 1.2.5.5
cyclic group 1.2.3.1
daughter phase 1.2.7.1
Deckoperation 1.2.2.1
decomposition of a group into cosets 1.2.4.2
detwinning 1.2.7.2
directional state of domain states 1.2.7.2
disorder–order phase transitions 1.2.7.3
distorted phase 1.2.7.1
domain boundary 1.2.7.1
domain region 1.2.7.1
domains
antiphase 1.2.7.3
ferroelectric 1.2.7.3
domain state 1.2.7.2, 1.2.7.2
directional state 1.2.7.2
symmetry state 1.2.7.2
domain structure 1.2.7
continuum description 1.2.7.1
microscopic description 1.2.7.1
translational 1.2.7.3
domain wall 1.2.7.1
enantiomorphic space groups 1.2.5.3, 1.2.6.2
enantiomorphic space-group types 1.2.5.3
Euclidean group 1.2.6.3, 1.2.6.3
Euclidean normalizer 1.2.6.3
specialized 1.2.6.3
typical 1.2.6.3
factor group 1.2.4.4
ferroelastic phase transitions 1.2.7.1
BaTiO3 1.2.7.3
ferroelectric domains 1.2.7.3
ferroelectric phase transitions
BaTiO3 1.2.7.3
fixed point 1.2.2.1, 1.2.2.1
gadolinium molybdate (Gd2(MoO4)3) 1.2.7.3
general subgroup 1.2.6.2
generators 1.2.3.1, 1.2.5.1
geometric crystal class 1.2.5.2
geometric mapping 1.2.2.1
glide plane 1.2.2.1
glide reflection 1.2.2.1
glide vector 1.2.2.1
group elements 1.2.3.1
group isomorphism 1.2.3.2
group postulates 1.2.3.1
groups 1.2.3
Abelian 1.2.3.1
abstract 1.2.3.2
commutative 1.2.3.1
cyclic 1.2.3.1
Euclidean 1.2.6.3, 1.2.6.3
factor 1.2.4.4
homomorphic 1.2.3.2
infinite 1.2.3.1
isomorphic 1.2.3.2, 1.2.3.2
orthogonal 1.2.6.3
plane 1.2.5.1
symmetry 1.2.3.1
translation 1.2.2.2, 1.2.5.1, 1.2.6.3
group–subgroup relations between space groups 1.2.6.1
group table 1.2.3.2
handedness 1.2.2.1, 1.2.2.1, 1.2.5.3
Hermann, theorem of 1.2.8.1, 1.2.8.1
Hermann's group 1.2.7.2
hexagonal crystal family 1.2.5.5
holohedral crystal class 1.2.5.4
holohedry 1.2.5.4, 1.2.5.4
homomorphic groups 1.2.3.2
homomorphic image (mapping) 1.2.3.2, 1.2.3.2, 1.2.4.4
homomorphism 1.2.3.2, 1.2.3.2, 1.2.4.4
kernel of 1.2.4.4
identity mapping (operation) 1.2.2.1, 1.2.3.1
image, homomorphic 1.2.3.2, 1.2.4.4
image point 1.2.2.1, 1.2.2.1
index of a subgroup 1.2.4.2, 1.2.6.2, 1.2.8.2
infinite group 1.2.3.1
invariant subgroup 1.2.4.2
inverse operation (element) 1.2.2.1, 1.2.3.1
inversion 1.2.2.1, 1.2.5.1
inversion centre 1.2.2.1
isometry 1.2.2.1, 1.2.2.5
isomorphic groups 1.2.3.2, 1.2.3.2
isomorphic subgroups 1.2.6.2, 1.2.8.2, 1.2.8.2
isomorphism 1.2.3.2
isomorphism class 1.2.3.2
isomorphism type 1.2.3.2
kernel
of a homomorphism 1.2.4.4
klassengleiche (k-) subgroups 1.2.6.2, 1.2.8.1
Lagrange, theorem of 1.2.4.2
lattice basis 1.2.2.2
lattice
Bravais 1.2.5.4
centred 1.2.5.1
primitive 1.2.5.1
lattice parameters 1.2.2.5
lattice system 1.2.5.2, 1.2.5.4
lattice type 1.2.5.4
lattice vector 1.2.2.2, 1.2.5.1
law, associative 1.2.3.1
law of composition 1.2.3, 1.2.3.1
left coset 1.2.4.2
length of a conjugacy class 1.2.4.3
length of a coset 1.2.4.2
linear part 1.2.2.3
macroscopic crystal 1.2.2.1
mapping
geometric 1.2.2.1
homomorphic 1.2.3.2, 1.2.4.4
identity 1.2.2.1
orthogonal 1.2.2.6
reversible 1.2.2.1
matrix
augmented 1.2.2.4
metric 1.2.2.5
transformation 1.2.2.7
matrix–column pair 1.2.2.3
matrix part 1.2.2.3
maximal subgroups 1.2.4.1, 1.2.6.2, 1.2.8.1, 1.2.8.2
metric matrix 1.2.2.5
microscopic description of domain structures 1.2.7.1
minimal supergroups 1.2.4.1
mirror plane 1.2.2.1
motion 1.2.2.1
rigid 1.2.2.1
multiplication table 1.2.3.2
neutral element 1.2.3.1
N-fold rotation 1.2.2.1
N-fold rotoinversion 1.2.2.1
non-ferroelastic phase transitions 1.2.7.1
non-isomorphic subgroups 1.2.6.2, 1.2.8.2
normalizers 1.2.4.5, 1.2.4.5, 1.2.6.3
affine 1.2.6.3
Euclidean 1.2.6.3
normal subgroups 1.2.4.2, 1.2.4.3
order–disorder phase transitions 1.2.7.3
order of a group 1.2.3.1
order of an element 1.2.3.1
orientational (directional) state of domain states 1.2.7.2
origin choice 1.2.5.1, 1.2.5.1
origin shift 1.2.2.7
orthogonal group 1.2.6.3
orthogonal mapping 1.2.2.6
parent clamping approximation (PCA) 1.2.7.1
parent phase 1.2.7.1
periodicity 1.2.2.1
perovskite, phase transitions 1.2.7.3
phase transitions
CuZn (β-brass) 1.2.7.3
ferroelastic 1.2.7.1, 1.2.7.3
ferroelectric 1.2.7.3
Gd2(MoO4)3 1.2.7.3
non-ferroelastic 1.2.7.1
order–disorder 1.2.7.3
plane group 1.2.5.1
plane-group types 1.2.5.3
point, fixed 1.2.2.1
point groups 1.2.5.4
crystallographic 1.2.5.4, 1.2.5.4
lattice 1.2.5.4
point-group type 1.2.5.4, 1.2.5.4
primitive basis 1.2.2.2
primitive lattice 1.2.5.1
product of group elements 1.2.3.1
product of sets of group elements 1.2.4.4
proper subgroups 1.2.4.1
prototypic phase 1.2.7.1
reduced basis 1.2.2.2
reflection 1.2.2.1
reflection plane 1.2.2.1
region of a domain 1.2.7.1
representatives, coset 1.2.4.2
reversible mapping 1.2.2.1
right coset 1.2.4.2
rigid motion 1.2.2.1
rotation 1.2.2.1, 1.2.2.1
N-fold 1.2.2.1
rotation axis 1.2.2.1
rotoinversion 1.2.2.1
N-fold 1.2.2.1
rotoinversion axis 1.2.2.1
screw axis 1.2.2.1
screw rotation 1.2.2.1
screw vector 1.2.2.1
Seitz symbol 1.2.2.3
set of generators 1.2.3.1
single-domain structure 1.2.7.2
space groups 1.2.5.1
classification of 1.2.5.2
enantiomorphic 1.2.5.3, 1.2.6.2
n-dimensional 1.2.5.1
subgroups of 1.2.8
symmorphic 1.2.5.3
space-group types 1.2.5.3
crystallographic 1.2.5.3
enantiomorphic 1.2.5.3
symmorphic 1.2.5.3
spontaneous strain components 1.2.7.1
subgroups 1.2.4.1
conjugate 1.2.4.3, 1.2.4.5
general 1.2.6.2
invariant 1.2.4.2
isomorphic 1.2.6.2, 1.2.8.2, 1.2.8.2
klassengleiche 1.2.6.2, 1.2.8.1
non-isomorphic 1.2.6.2, 1.2.8.2
of space groups 1.2.6, 1.2.8
proper 1.2.4.1
translationengleiche 1.2.6.2, 1.2.8.1, 1.2.8.2
trivial 1.2.4.3
supergroups 1.2.4.1
minimal 1.2.4.1
symbols
Seitz 1.2.2.3
symmetry
centre of 1.2.2.1
symmetry element 1.2.3.1
symmetry group 1.2.3.1
symmetry operation 1.2.2.1
crystallographic 1.2.2.1, 1.2.3
symmetry state of domain states 1.2.7.2
symmorphic space groups 1.2.5.3
symmorphic space-group types 1.2.5.3
theorems
Hermann's theorem 1.2.8.1, 1.2.8.1
Lagrange's theorem 1.2.4.2
transformation matrix 1.2.2.7
transformation of bases and coordinates 1.2.2.7, 1.2.2.7
transformation of coordinate system 1.2.2.7
transformation twin 1.2.7.1
translation 1.2.2.1, 1.2.5.1, 1.2.5.1
translational domain structure 1.2.7.3
translationengleiche (t-) subgroups 1.2.6.2, 1.2.8.1, 1.2.8.2
translation group 1.2.2.2, 1.2.5.1, 1.2.6.3
translation part 1.2.2.3
translation twins 1.2.7.3
translation vector 1.2.2.1, 1.2.2.2, 1.2.5.1
trivial subgroup 1.2.4.3
twinning, twins 1.2.7.1
unit element 1.2.3.1
vector 1.2.2.6
lattice 1.2.2.2, 1.2.5.1
translation 1.2.2.1, 1.2.2.2, 1.2.5.1
vector coefficients 1.2.2.2, 1.2.2.6
vector lattice 1.2.2.2, 1.2.5.1