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Symmetry group of a configuration in space

In mathematics, physics and chemistry, a space group is the symmetry group of a repeating pattern in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of the pattern that leave it unchanged. In three dimensions, space groups are classified into 219 distinct types, or 230 types if chiral copies are considered distinct. Space groups are discrete cocompact groups of isometries of an oriented Euclidean space in any number of dimensions. In dimensions other than 3, they are sometimes called Bieberbach groups. In crystallography, space groups are also called the crystallographic or Fedorov groups, and represent a description of the symmetry of the crystal. A definitive source regarding 3 dimensional space groups is the International Tables for Crystallography .

Space groups in 2 dimensions are the 17 wallpaper groups which have been known for several centuries, though the proof that the list was complete was only given in 1891, after the much more difficult classification of space groups had largely been completed. In 1879 the German mathematician Leonhard Sohncke listed the 65 space groups (called Sohncke groups) whose elements preserve the chirality. More accurately, he listed 66 groups, but both the Russian mathematician and crystallographer Evgraf Fedorov and the German mathematician Arthur Moritz Schoenflies noticed that two of them were really the same. The space groups in three dimensions were first enumerated in 1891 by Fedorov (whose list had two omissions (I3d and Fdd2) and one duplication (Fmm2)), and shortly afterwards in 1891 were independently enumerated by Schönflies (whose list had four omissions (I3d, Pc, Cc, ?) and one duplication (P21m)). The correct list of 230 space groups was found by 1892 during correspondence between Fedorov and Schönflies. later enumerated the groups with a different method, but omitted four groups (Fdd2, I2d, P21d, and P21c) even though he already had the correct list of 230 groups from Fedorov and Schönflies; the common claim that Barlow was unaware of their work is incorrect. describes the history of the discovery of the space groups in detail.

The space groups in three dimensions are made from combinations of the 32 crystallographic point groups with the 14 Bravais lattices, each of the latter belonging to one of 7 lattice systems. What this means is that the action of any element of a given space group can be expressed as the action of an element of the appropriate point group followed optionally by a translation. A space group is thus some combination of the translational symmetry of a unit cell (including lattice centering), the point group symmetry operations of reflection, rotation and improper rotation (also called rotoinversion), and the screw axis and glide plane symmetry operations. The combination of all these symmetry operations results in a total of 230 different space groups describing all possible crystal symmetries. The number of replicates of the asymmetric unit in a unit cell is thus the number of lattice points in the cell times the order of the point group. This ranges from 1 in the case of space group P1 to 192 for a space group like Fmm, the NaCl structure.

The elements of the space group fixing a point of space are the identity element, reflections, rotations and improper rotations, including inversion points.

The translations form a normal abelian subgroup of rank 3, called the Bravais lattice (so named after French physicist Auguste Bravais). There are 14 possible types of Bravais lattice. The quotient of the space group by the Bravais lattice is a finite group which is one of the 32 possible point groups.

A glide plane is a reflection in a plane, followed by a translation parallel with that plane. This is noted by , , or , depending on which axis the glide is along. There is also the glide, which is a glide along the half of a diagonal of a face, and the glide, which is a fourth of the way along either a face or space diagonal of the unit cell. The latter is called the diamond glide plane as it features in the diamond structure. In 17 space groups, due to the centering of the cell, the glides occur in two perpendicular directions simultaneously, i. e. the same glide plane can be called b or c, a or b, a or c. For example, group Abm2 could be also called Acm2, group Ccca could be called Cccb. In 1992, it was suggested to use symbol e for such planes. The symbols for five space groups have been modified:
Space group no. 39 41 64 67 68 New symbol Aem2 Aea2 Cmce Cmme Ccce Old Symbol Abm2 Aba2 Cmca Cmma Ccca

A screw axis is a rotation about an axis, followed by a translation along the direction of the axis. These are noted by a number, n, to describe the degree of rotation, where the number is how many operations must be applied to complete a full rotation (e. g., 3 would mean a rotation one third of the way around the axis each time). The degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. So, 21 is a twofold rotation followed by a translation of 1/2 of the lattice vector.

The 65 "Sohncke" space groups, not containing any mirrors, inversion points, improper rotations or glide planes, yield chiral crystals, not identical to their mirror image; whereas space groups that do include at least one of those give achiral crystals. Achiral molecules sometimes form chiral crystals, but chiral molecules always form chiral crystals, in one of the space groups that permit this. Among the 65 Sohncke groups are 22 that come in 11 enantiomorphic pairs.

Only certain combinations of symmetry elements are possible in a space group. Translations are always present, and the space group P1 has only translations and the identity element. The presence of mirrors implies glide planes as well, and the presence of rotation axes implies screw axes as well, but the converses are not true. An inversion and a mirror implies two fold screw axes, and so on.

In n dimensions, an affine space group, or Bieberbach group, is a discrete subgroup of isometries of n dimensional Euclidean space with a compact fundamental domain. proved that the subgroup of translations of any such group contains n linearly independent translations, and is a free abelian subgroup of finite index, and is also the unique maximal normal abelian subgroup. He also showed that in any dimension n there are only a finite number of possibilities for the isomorphism class of the underlying group of a space group, and moreover the action of the group on Euclidean space is unique up to conjugation by affine transformations. This answers part of Hilbert's eighteenth problem. showed that conversely any group that is the extension of Zn by a finite group acting faithfully is an affine space group. Combining these results shows that classifying space groups in n dimensions up to conjugation by affine transformations is essentially the same as classifying isomorphism classes for groups that are extensions of Zn by a finite group acting faithfully. It is essential in Bieberbach's theorems to assume that the group acts as isometries; the theorems do not generalize to discrete cocompact groups of affine transformations of Euclidean space. A counter example is given by the 3 dimensional Heisenberg group of the integers acting by translations on the Heisenberg group of the reals, identified with 3 dimensional Euclidean space. This is a discrete cocompact group of affine transformations of space, but does not contain a subgroup Z3.

№ (Space group number) Crystal system, (count), Bravais lattice Point group Space groups (international short symbol) Int'l Schön. Orbifold Cox. Ord. 1Triclinic(2) 1 C111[ ]+1 P1 2 Ci1×[2+,2+]2P 3–5Monoclinic(13) 2 C2 22 [2]+ 2P2, P21C2 6–9 m Cs *11 [ ]2Pm, PcCm, Cc 10–15 2/m C2h 2* [2,2+]4P2/m, P21/mC2/m, P2/c, P21/cC2/c 16–24Orthorhombic(59) 222 D2 222 [2,2]+4P222, P2221, P21212, P212121, C2221, C222, F222, I222, I212121 25–46 mm2 C2v *22 [2]4Pmm2, Pmc21, Pcc2, Pma2, Pca21, Pnc2, Pmn21, Pba2, Pna21, Pnn2Cmm2, Cmc21, Ccc2, Amm2, Aem2, Ama2, Aea2Fmm2, Fdd2Imm2, Iba2, Ima2 47–74 mmm D2h *222 [2,2]8Pmmm, Pnnn, Pccm, Pban, Pmma, Pnna, Pmna, Pcca, Pbam, Pccn, Pbcm, Pnnm, Pmmn, Pbcn, Pbca, PnmaCmcm, Cmce, Cmmm, Cccm, Cmme, CcceFmmm, FdddImmm, Ibam, Ibca, Imma 75–80 Tetragonal(68) 4 C444[4]+4P4, P41, P42, P43, I4, I41 81–82 S42×[2+,4+]4P, I 83–88 4/m C4h4*[2,4+]8P4/m, P42/m, P4/n, P42/nI4/m, I41/a 89–98 422 D4224[2,4]+8P422, P4212, P4122, P41212, P4222, P42212, P4322, P43212I422, I4122 99–110 4mm C4v*44[4]8P4mm, P4bm, P42cm, P42nm, P4cc, P4nc, P42mc, P42bcI4mm, I4cm, I41md, I41cd 111–122 2m D2d2*2[2+,4]8P2m, P2c, P21m, P21c, Pm2, Pc2, Pb2, Pn2Im2, Ic2, I2m, I2d 123–142 4/mmm D4h*224[2,4]16P4/mmm, P4/mcc, P4/nbm, P4/nnc, P4/mbm, P4/mnc, P4/nmm, P4/ncc, P42/mmc, P42/mcm, P42/nbc, P42/nnm, P42/mbc, P42/mnm, P42/nmc, P42/ncmI4/mmm, I4/mcm, I41/amd, I41/acd 143–146Trigonal(25) 3 C333[3]+3P3, P31, P32R3 147–148 S63×[2+,6+]6P, R 149–155 32 D3223[2,3]+6P312, P321, P3112, P3121, P3212, P3221R32 156–161 3m C3v*33[3]6P3m1, P31m, P3c1, P31cR3m, R3c 162–167 m D3d2*3[2+,6]12P1m, P1c, Pm1, Pc1Rm, Rc 168–173 Hexagonal(27) 6 C666[6]+6P6, P61, P65, P62, P64, P63 174 C3h3*[2,3+]6P 175–176 6/m C6h6*[2,6+]12P6/m, P63/m 177–182 622 D6226[2,6]+12P622, P6122, P6522, P6222, P6422, P6322 183–186 6mm C6v*66[6]12P6mm, P6cc, P63cm, P63mc 187–190 m2 D3h*223[2,3]12Pm2, Pc2, P2m, P2c 191–194 6/mmm D6h*226[2,6]24P6/mmm, P6/mcc, P63/mcm, P63/mmc 195–199Cubic(36) 23 T332[3,3]+12P23, F23, I23P213, I213 200–206 m Th 3*2[3+,4]24Pm, Pn, Fm, Fd, Im, Pa, Ia 207–214 432 O432[3,4]+24P432, P4232F432, F4132I432P4332, P4132, I4132 215–220 3m Td *332[3,3]24P3m, F3m, I3mP3n, F3c, I3d 221–230 mm Oh *432[3,4]48Pmm, Pnn, Pmn, PnmFmm, Fmc, Fdm, FdcImm, Iad

Note: An e plane is a double glide plane, one having glides in two different directions. They are found in seven orthorhombic, five tetragonal and five cubic space groups, all with centered lattice. The use of the symbol e became official with
The lattice system can be found as follows. If the crystal system is not trigonal then the lattice system is of the same type. If the crystal system is trigonal, then the lattice system is hexagonal unless the space group is one of the seven in the rhombohedral lattice system consisting of the 7 trigonal space groups in the table above whose name begins with R. (The term rhombohedral system is also sometimes used as an alternative name for the whole trigonal system.) The hexagonal lattice system is larger than the hexagonal crystal system, and consists of the hexagonal crystal system together with the 18 groups of the trigonal crystal system other than the seven whose names begin with R.

The Bravais lattice of the space group is determined by the lattice system together with the initial letter of its name, which for the non rhombohedral groups is P, I, F, A or C, standing for the principal, body centered, face centered, A face centered or C face centered lattices. There are seven rhombohedral space groups, with initial letter R.