Proper affine class of one-dimensional magnetic space groups T11.1.1
Proper arithmetic class of integral matrix groups 82
Proper congruence class of groups of isometries T4.2
Proper congruence class of point sets 190 N7.2
Proper holohedral class of groups of rotations 216
Proper holohedral class of matrix groups 239 241
Proper motion = direct isometry 121
Proper O-equivalence class of matrix groups 81
Proper rotation 116 121 148 162
Proper similarity class of groups of isometries T4.2
Proper similarity class of point sets 190 N7.2
Proper space-group type = proper affine class of space groups 264
Proper subarithmetic class of Bravais groups of magnetic point lattices 424
Proper subgroup 15
Proper subset 3
Proximity cell N8.8
Q-symmetry groups 360
Quadratic form 96
Quasi-crystal Npr
Quasi-H-M symbol of a spin point group 510 T20.1
Quasisymmetry groups 360
Quotient group = factor group 19 21
Quotient set 7
r-equivalence class of (3+1)-dimensional Bravais superspace groups 553
r-equivalence class of (3+1)-dimensional superspace groups = superspace-group types 557
R-equivalent elements of a set 7 8
R-subgroup of a space group 260 436 437 449 453
R-supergroup of a space group 260
Rational angle 161
Rational discrete helix 336
Rational helical spin arrangement 489
Ray 521
Real crystal 293
Real linear group 103
Real orthogonal matrix group 79 109
Real vector space 88
Realization of an abstract group 14
Reciprocal bases 211
Reciprocal point lattice associated with a point lattice 211 N8.3
Rectangular Bravais system of point lattices T8.2
Rectangular Bravais system of space groups T9.1.3
Rectangular geometric-class system of groups of rotations T6.7
Rectangular groups of rotations T6.7
Rectangular ITC-system of space groups T9.1.3
Rectangular point lattice, centred 224 234
Rectangular point lattice, primitive 224 234
Reduced basis of a point lattice in the sense of Dirichlet 207 212 N8.5
Reduced basis of a point lattice in the sense of Minkowski N8.5
Reduced colour-group family of a group of isometries 370 T13.1
Reduced family of a group of proper rotations 157 T6.4
Reduced magnetic families of crystallographic magnetic groups of rotations T15.3 T15.4
Reduced magnetic family of a group of isometries 405
Reduced magnetic family of a lattice group 429
Reduced magnetic superfamily of a group of isometries 405
Reduced matrices of the same form 132
Reduced matrix 132
Reduced metric form of a point lattice in the sense of Dirichlet 212 N8.5
Reduced metric form of a point lattice in the sense of Minkowski N8.5
Reduced net-group family of a two-dimensional space group 310
Reduced net-group superfamily of a two-dimensional space group 310
Reduced spin-group family of a group of isometries 515
Reduced spin-group family of a group of isometries and of a group of orthogonal transformations 515
Reduced spin-group family of the group 422 T20.1
Reduced superfamily of a group of proper rotations 157
Reduced unit cell of a point lattice 212 213
Reducible group of rotations 159 160
Reducible matrix group 132 135
Reducible representation of a group 135
Reference plane 330
Referring a group of affine transformations to a coordinate system 122
Referring a group of linear transformations to a basis 106 107
Reflection group in space 151
Reflection group of a plane 163
Reflection in a line 163
Reflection in a plane 150
Reflexive relation 6
Regular matrix representation of a group 49
Regular permutation representation of a group 49 70
Relabelling of a bracket symbol of a permutation 35
Relabelling Theorem for Permutations 35
Relation on a set 6 8
Relation on a set, equivalence 6
Relation on a set, reflexive 6
Relation on a set, symmetric 6
Relation on a set, transitive 6
Relativistic crystallographic groups N7.12
Representation of a group by a group of linear transformations 131 N5.1
Representation of a group by a group of matrices = matrix representation of a group 24 131
Representation of a group by a group of permutations = permutation representation of a group 24
Representation of a matrix group by itself 131
Representation of an affine group by an inhomogeneous linear group 113
Representation space = carrier space of a representation 131
Representations (one-dimensional) of the space-time inversion group T14.1
Representative of a left coset 19
Representative of a right coset 19
Restricted arithmetic class = rZ-class of restricted Bravais groups 552
Restricted Bravais group of a (3+1)-dimensional point lattice associated with a Bravais group of a three-dimensional point lattice 550
Restriction of a homomorphism of a group to its subgroup 24
Reversal space groups in two dimensions N12.4
Rhombohedral Bravais system = trigonal Bravais system
Rhombohedral ITC-system = trigonal fTC-system
Rhombohedral point lattice = trigonal point lattice
Right cosets of a subgroup 8 19
Right-handed coordinate systems 120
Rigid motion = isometry 115 N4.4
Rotation = proper or improper rotation 148 149
Rotation at a point of a Euclidean point space 116
Rotation group associated with a space group 250
Rotation matrix 109
Rotation-matrix group associated with a basis 109
Rotational part of an isometry 117 N4.4
Rotatory inversion = rotatory reflection 151
Rotatory reflection 151
Rule for constructing the h-subgroups 64 515
Rule for finding all representations by transitive groups of permutations 74
scalar 88
scalar functions 387 389
Scalar product (= inner product) of two vectors 95
Schoenflies symbols for the groups of rotations 158
Schoenflies symbols for the space groups 279
Schroedinger equation 522
Schur's lemma 141 N5.3
Schwarz's inequality 95
Screw axis 152
Screw displacement 152 259
Self-conjugate subgroup = normal subgroup 26
Semidirect product group 27 52
Semidirect product of  and a group of its automorphisms 78 524
Semidirect product of two groups 52 53
Semidirect product of two groups, external 52 56
Semidirect product of two groups, internal 56
Semilinear group of a complex vector space 523
Semilinear transformation 519 523
Senior group of symmetry and antisymmetry 361
Sesquilinear transformation of a complex vector space 523
Set 3
Set associated with a partition of a set 6
Set generated by a group of permutations of a set 38
Set of generating elements (= generators) of a subgroup 17
Set of point lattices associated with a class of positive definite symmetric matrices 209
Set of positive definite symmetric matrices associated with an integral-matrix group 225
Setting of a space group 279
Shifting the origin of a coordinate system 94
Shortest lattice vectors of an n-dimensional point lattice 213
Shubnikov group 358 406
Similar groups of isometries T4.2
Similar point sets 190 N7.2
Similarity class of groups of isometries T4.2
Similarity class of point sets 190 N7.2
Similarity group of a Euclidean point space 117
| Similarity group of a Euclidean vector space 117
Similarity transformation of a Euclidean point space 117
Simple crystal 196 290 T10.2
Simple group 21 24 Npr
Simple point set 189
Simple point set having some symmetry 192
Simple set 39
Simple spin arrangement 478 482 486
Singleton 3
Site group for a group of isometries 194
Site group of a point 194
Site group of a point (= of an atom site) of a simple crystal 290
Site point group of a point (= of an atom site) of a simple crystal 290
Site space group of a point (= of an atom site) of a simple crystal 291 292
Site symmetry group of a point 194
Space = ordinary space = three-dimensional Euclidean point space 87 101 148
Space group 17 46 187 195 218 219 250
Space group as an extension of a lattice group 256 N9.4
Space groups of a plane T9.1.3
Space inversion = inversion
Space of functions = function space
Space-and-time inversion 166
Space-group type = affine class of space groups 264
Space-only spin group 506
Space-time 8 87 99 101 N3.1
Space-time inversion group 13 166
Space-time orientation preserving group 166
Space-time rotation group 166
Space-time translation group 166
Special element of a set 39
Special inhomogeneous linear group SIL(n) 119
Special inhomogeneous rotation-matrix group associated with a basis 116
Special integral matrix group SL(n, Z) 79
Special linear real matrix group SL(n, R) 79
Special orthogonal-matrix group SO(n) 79 110
Special position of a point 194 304
Special rational matrix group SL(n, Q) 79
Special rotation-matrix group associated with a basis 110
Special unitary matrix group SU(n) 77
Spin = magnetic dipole moment 400
Spin arrangement = magnetic dipole arrangement 356 419 420
Spin arrangement generated by a magnetic lattice group 420
Spin arrangement generated by a magnetic space group 479
Spin group = vector group 354 359 505 N12.6
Spin group of rotations = spin point group 508 515
Spin lattice 420
Spin lattice generated by a lattice group 420
Spin lattice group = spin translation group 508 517
Spin point group = spin group of rotations 508
Spin space 400 509 510
Spin space group 508 510 517 N12.6
Spin translation group = spin lattice group 508
Spin-group classification label of a spin arrangement 510
Spin-group family of a group of isometries 508
Spin-group family of a group of isometries and of a group of orthogonal transformations 508
Spin-only spin group 506
Split extension of a group 52
Square Bravais system of point lattices T8.2
Square Bravais system of space groups T9.1.3
Square geometric-class system of groups of rotations T6.7
Square groups of rotations T6.7
Square ITC-system of space groups T9.1.3
Square matrix =  matrix 13
Square point lattice 234
Stabilizer of a point of a Euclidean point space 116
Stabilizer of a point of an affine point space 111
Stabilizer of an element of a set 38
Standard basis of a (1+1)-dimensional point lattice 543
Standard basis of a (3+1)-dimensional point lattice 547 T22.1
Standard basis of a coordinate space 90
Standard basis of a point lattice 235 T8.2 T8.3
Standard basis of, irrational, a (3+1)-dimensional point lattice 550
Standard representation of a discrete group of isometries 476
Standard representation of a magnetic group 476
Standard representation of a magnetic space group 476
Standard representation of the point group of a magnetic space group 477
Standard representation of the point group of a space group 477
Standard rule for constructing  -subgroups 61
Standard rule for constructing a colour function 367 376
Standard rule for constructing net groups 309
Standard rule for constructing spin arrangements 479
Standard rule for constructing the magnetic family of a group of isometries 404
Standard set of coset representatives for a space group 281
State vector 522
Stationary state belonging to the energy eigenvalue 522
Stratum of orbits for a group of isometries 295
Stretching factor 117
Strictly  -equivalent matrix groups 81
Strictly  -equivalent matrix groups 78
Strictly  -equivalent matrix groups 81
Strictly congruent bases of a Euclidean vector space 110 120
Strictly congruent coordinate systems in a Euclidean point space 120
Strictly congruent groups of isometries T4.2
Strictly congruent point sets 190
Strictly equivalent groups of isometries 122 T4.2
Strictly equivalent magnetic groups 405
Strictly equivalent net groups 310
Strictly O-equivalent matrix groups 81
Strictly similar groups of isometries T4.2
Strictly similar point sets 190
Structure of a group 14
Subarithmetic class of Bravais groups of magnetic point lattices 424
Subdirect product associated with two quotient groups 67 320
Subdirect product group associated with a group of proper rotations 174
Subdirect product of two groups 59
Subduced matrix representation of a subgroup = restriction to a subgroup of a group of the latter's homomorphism onto a matrix group 24
Subducing a group of permutations of a subset by a group of permutations of a set 38 191
Subfield 52
Subgroup criterion 16
Subgroup of a group 15
Subgroup of a group generated by a set of elements of the group 17
Subgroup of a group generated by a set of subgroups of a group 18
Subgroup of index 2 29
Subgroup of the space-time rotation group = group of rotations of space-time 169
Subgroups of the direct product of two groups 59
Sublattice coholohedral with a magnetic point lattice 422
Sublattice coholohedral with a point lattice 422
Sublattice of a point lattice 204
Subperiodic group of isometries 196
Subset 3
Subset of a group 16
Subspace (of a vector space) spanned by a set of linearly independent vectors 89
Sum of two vectors 88
Superconducting medium 397
Supercrystal ((3+1)-dimensional) 548
Superfamily of a group 61
Superfamily of a group of proper rotations 157
Supergroup of a group 16
Superspace ((1+1)-dimensional) 544
Superspace ((3+1)-dimensional) 545
Superspace ((3+d)-dimensional) 538
Superspace group 197 199 537
Superspace group ((1+1)-dimensional) 544
Superspace group ((3+1)-dimensional) 547 556 557 T22.2
Superspace group ((3+d)-dimensional) 537
Superspace-group type 557
Surjection 5
Symmetric group of degree n 33
Symmetric group on a set 12 30 32 45
Symmetric matrix 76
Symmetric quadratic form 96
Symmetric relation 6
Symmetry class of point sets 193
Symmetry colour group of a colour function 364
Symmetry elements 37
Symmetry elements of a point set 191. N7.3
Symmetry group and symmetry elements of a subset 37
Symmetry group of a composite crystal 300
Symmetry group of a function in the sense of an action of a group 44
Symmetry group of a line 161
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