Seven space groups cannot make their own translations

Six groups are made of glides asked which plane groups are generated by their own reflections, counting the lattice translations as something the reflections must produce rather than something handed to them. It found that glides generate six of the seventeen and mirrors four, that together they generate eleven, and that two groups, p1 and pm, cannot be made from their own operations at all: each needs a translation given outright. It ended with the same question in space. Three on a plane, and never four had meanwhile left a census over the whole list of space groups undone, and this is one.

The answer, over all 219 affine types of space group, is almost uniform. Two hundred and twelve are generated by their own operations; seven are not. The seven are P1, P2, Pm, Pmm2, P3, P4 and P6. They are exactly the seven plane groups that have no glide, p1, p2, pm, pmm, p3, p4 and p6, each repeated along a third direction that every one of its operations fixes. Nothing in such a group moves along that direction except the translation itself. That the list is so short, when every polar crystal class has groups of this shape, comes from a difference between the plane and space that the plane essay could not see. A glide line can only slide along itself. A glide plane can slide in any direction within it, including up an axis.

Building all 219 #

The census needs every space-group type as a list of operations with their translation parts, not as a symbol. Two hundred and nineteen over seventy-three built exactly that. For each of the seventy-three arithmetic classes it computed the group of translation classes $H^1$ and its orbits under the normaliser, and each orbit is one type. Taking one representative class from each orbit and attaching its translations to the point group’s matrices gives the 219 groups as explicit operations, each on a primitive basis of its lattice. The census works from those, without consulting a table.

Each operation is then sorted by kind, and the sort has to be done operation by operation, not coset by coset. An operation of order $n$ with determinant one is a screw when its screw component, the average of its translation over the $n$ powers, is not a lattice vector, and a rotation when that component is zero. A reflection is a glide when its slide, the component of its translation within its plane, is not a lattice vector, and a mirror when the slide is zero. Inversions and rotoinversions each fix a point, and they join the rotations and mirrors as point operations. An operation whose screw component or slide is a whole lattice vector is a point operation followed by a translation, and it is used as neither kind, since using it would hand the generators a lattice translation for free. Each operation is taken together with its translates by lattice vectors in a small box, because the same coset holds a mirror on one plane and a glide on the next.

Deciding whether a set generates the group #

A set of operations generates the group when its products reach every coset of the lattice and the translations among its products are the whole lattice. The first half is a search: follow products outward from the identity and record which cosets are reached. The second half is where a careless count goes wrong, and it has an exact answer in Schreier’s lemma.

The search leaves one chosen element of the generated subgroup in each coset it reaches. Multiply any generator by any chosen element, and compare the result with the chosen element of the coset it lands in. The two differ by a pure translation, and the lemma says that these translations, one for every generator and every chosen element, generate all the translations the subgroup contains. So the translations a set can make are computed completely from a finite list, not estimated from words of bounded length. Their span in the integer lattice is found by reducing them to echelon form. The set generates the whole group exactly when every coset is reached and the product of the echelon form’s diagonal, the index of the span, is one.

The census #

The picture at the head of this essay is the division. Screws and glides alone generate 177 of the 219 types. Rotations, mirrors, inversions and rotoinversions alone generate 139. Together they generate 212. The groups divide further. 110 are made by either kind on its own. 67 are made only by their screws and glides and 29 only by their point operations. Six need both kinds together, as cm, pmg and cmm did in the plane.

The first number is the surprising one. It says that the non-symmorphic operations, the screws and glides that a symmorphic group need not have at all, generate far more groups than the point operations do. Even among the seventy-three symmorphic types, forty-two are generated by their screws and glides alone. A symmorphic group has an origin at which every operation’s translation vanishes. But its centring translations, composed with its rotations and mirrors, produce screws and glides on other axes and planes, and those make the whole group.

A screw done twice climbs its axis #

The reason is the one that separated glides from mirrors in the plane, and space has it twice over, since eleven screws and no others give the rotations their own sliding versions. A mirror done twice is nothing and a glide done twice is a translation along its line. In the same way a rotation done $n$ times is nothing and a screw done $n$ times is a translation along its axis. $P2_1$ shows it at its simplest. Its two-fold screw, done twice, climbs one lattice step up its axis. The product of two parallel screws on different axes is a translation across them. Together those span the lattice, so $P2_1$ is made of its screws. P2 has the same arrangement of axes with rotations in place of screws. Its rotations done twice are nothing, and products of parallel rotations make translations across the axes and nothing along them, so no product of P2’s operations climbs the axis. P2 is one of the seven.

That is pg against pm one dimension up, and it is the whole mechanism of the census. Groups with screws or glides in enough directions make their own lattices. Groups without them have to be given the translations their point operations cannot produce.

The seven, and the plane groups they repeat #

The seven share a shape. Each has a primitive lattice and no screws or glides. Each point group is one of the ten polar classes, so every operation fixes one direction, the axis of the rotations or a line within the mirror. And each is a plane group repeated along that direction: P2 is p2 stacked straight up its axes, Pmm2 is pmm stacked up the line both mirrors contain. Nothing in any of the seven moves along the fixed direction, since rotations about it, mirrors containing it and translations across it all keep a point’s height. So the translation along it is not a product of anything else and has to be given.

The list of plane groups on the right is exactly the list of plane groups with no glide. Every other plane group has one, and the reason is precise. Repeating a plane group with a glide along a third direction does not give a group the census fails. In space, that glide becomes a glide plane containing the third direction, and a glide plane can slide along it.

A glide plane can slide up the axis #

P4mm is polar in the same way as P4. Every operation fixes the four-fold axis, and its rotations and mirrors alone make no translation along it. The census finds that its point operations reach every coset and span only a lattice of rank two. Yet P4mm is made of its own operations, and the reason is a glide plane. The diagonal mirror composed with a translation of one lattice step along the other axis is a glide in the diagonal plane. Its slide is half the face diagonal, $(\tfrac12, \tfrac12, 0)$. Composing the same mirror with a lattice step along the face and one up the axis gives another operation of the group, a glide in the same plane with slide $(\tfrac12, \tfrac12, 1)$. Squared, it is a translation by $(1, 1, 2)$, which climbs the axis. With that, the group’s own operations reach the whole lattice.

A glide line in the plane can never do this. Its slide lies along the line, and the lattice vectors along one line are the multiples of one vector, so every glide on it slides the same way. A line in the plane carries one kind of glide for the same reason that a line carries one screw in space. A glide plane contains a two-dimensional lattice, which is how a plane can carry two glides. Its slides can be half a vector in one direction plus a whole vector in another, and so a glide plane parallel to a polar axis supplies translations along that axis. The seven groups the census fails are exactly those with no glide plane parallel to the fixed direction, which means no glide in the plane group being repeated.

The centred polar groups escape for a related reason, the one by which Cm is not Pm. In Cm, I4 or R3 the centring translation has a component along the polar direction. Composed with a mirror or a rotation it produces a glide or a screw that climbs. Every centred polar group in the census is made of its own operations.

Where the groups made of screws and glides are #

By crystal family, the screws and glides do most of the work everywhere except in the triclinic and monoclinic families, where there are few screws to go round. The two triclinic types have none at all. The screws and glides generate five of the thirteen monoclinic types, forty-seven of the fifty-nine orthorhombic ones, fifty-three of the sixty-five tetragonal ones and thirty-eight of the forty-five in the hexagonal family. In the cubic family, thirty-four of the thirty-five types are generated by their screws and glides and thirty-four by their point operations, and every one of the thirty-five by one kind or the other. Its three-fold axes along the body diagonals turn any direction into two others, so a translation made along one axis is made along all three.

The seven groups that need a translation given are spread across the other families: one triclinic, two monoclinic, one orthorhombic, one tetragonal and two in the hexagonal family. The cubic family has none, because it has no polar class. The distribution is what the explanation predicts. A group can only fail if every operation fixes a common direction, which is the definition of a polar class.

Sixteen groups with no screw or glide, and the nine that escape #

Being polar is necessary as well as primitive and glide-free, and the census shows why. Sixteen of the 219 types have no screw and no glide anywhere among their operations. Seven of them are the seven. The other nine are $P\bar{1}$, P2/m, Pmmm, P222, P4/m, $P\bar{4}$, $P\bar{6}$, P6/m and $P\bar{3}$, and every one is generated by its point operations alone. What they share is an operation that reverses the direction the seven leave fixed.

$P\bar{1}$ is the simplest case. Its operations are inversions through the lattice points and through the half-lattice points between them. Two inversions through centres $c_1$ and $c_2$ compose to a translation by $2(c_2 - c_1)$. Taking centres half a lattice vector apart in each of the three directions makes all three lattice vectors. A lattice of inversion centres makes its own lattice, as a row of parallel mirrors makes the translations across it. P2/m adds to P2 a mirror perpendicular to the two-fold axis, and mirrors at heights $0$ and $\tfrac12$ compose to a step up the axis that no rotation could supply. P222 does it with two-fold axes lying across the principal axis, which turn the axis over. $P\bar{4}$ and $P\bar{6}$ do it with rotoinversions, since a four-fold rotoinversion carries the axis to its own negative. In each case a pair of operations that reverse the axis at different heights composes to a climb.

So the seven are exactly the groups caught by three conditions at once: no glide or screw to slide along the fixed direction, no centring translation with a component along it, and no operation that turns it over. Remove any one and the group makes its own lattice. The census does not have to be told this. It is what the list of seven says when set beside the list of sixteen.

Six groups that need both kinds #

The six types generated only by screws and glides and point operations together are C2, Cm, Cmm2, Amm2 and P4mm, all symmorphic, and one non-symmorphic type in the arithmetic class Pmm2. In each, one kind on its own falls short of the lattice in a single direction, and the other kind makes up exactly that direction. In P4mm the point operations reach all eight cosets and make translations across the axis only. The glides alone reach only four of the cosets, the half of the point group that contains no four-fold rotation, and the translations among their products span a sublattice of index two. Neither is the group. Together they are.

The same thing happened in the plane with cm, pmg and cmm, set beside the four groups made of mirrors, and there too it was decided by the centring or by a glide with a slide in the one direction the mirrors could not reach. Space has no new mechanism here. What it adds is the glide plane’s second direction of slide, which lets P4mm into this list where P4 stays among the seven.

What the census rests on #

The census depends on the representatives the orbit computation chose. Any member of an orbit is the same space-group type, conjugate by a change of basis, and conjugation preserves both which cosets a set reaches and the index of the lattice it spans. So the choice cannot change a result. It also depends on counting individual operations with their translates, not cosets. That is what lets P4mm’s glide with a step up the axis into the generating set, and a census working modulo the lattice would have missed it in the other direction as well. It would have called P2 generated, by handing its rotations the lattice translation they cannot make.

Nor does the census use the International Tables’ symbols. The seven groups are named here by arithmetic class, and within each class the one that fails is the symmorphic one. That can be read off the representative without a symbol: its translation class is zero. The plane groups beside them are the projections along the fixed direction, which is how the correspondence can be checked by hand.

What the census has to refuse #

The refused claim is the tempting generalisation of the seven: that a polar space group can never make the translation along its polar axis. Cm, I4, R3, P4mm, P3m1, P31m and P6mm are all polar and all made of their own operations. The first three succeed through their centring and the last four through their glide planes. The test that guards the method is the lattice index on known cases: a sublattice of index two, a centred sublattice and a set of rank two, each computed correctly. The reduction that decides the whole census is the one step that could fail silently.

Still open: the smallest generating sets in space #

In the plane two glides made each of five groups and p3m1 needed three. The corresponding count in space asks: for each of the 212 groups that can be generated, how few of its own operations suffice, and of which kinds? Two operations generate a group only if their products reach every coset, and space groups have up to forty-eight cosets, so some will need three or four. The Schreier test makes each candidate set cheap to check. The search over sets is not, and it has not been run. The count would say which space groups are two-generated. That is a question about the groups as abstract groups as much as about their geometry, and the International Tables’ lists of generators do not answer it.