The richest group has the poorest arithmetic

The counts so far have been rows of indices — the sizes a group’s cell can grow by while the group comes back the same. The row is a product over the primes, and the rows get shorter as the point group gets larger: p1 has seven hundred and sixty-two invariant sublattices between index two and index thirty, and p4m, p4g and p6m have eight each.

That trend is easy to read as a curiosity about the plane. It is not; it is the whole subject, and the place it is clearest is the group with the most symmetry there is.

Seven indices in forty #

A cubic lattice has a great many sublattices. At index sixteen there are six hundred and fifty-one; at index thirty there are hundreds more. The full cubic point group — forty-eight operations, the rotations of a cube and their products with the inversion — carries almost none of them to itself.

To index forty there are seven: at 1, 2, 4, 8, 16, 27 and 32. And there is exactly one at each.

The shapes name themselves. Scaling the whole lattice by a whole number m multiplies the index by $m^3$ and gives the primitive cubic lattice again; that is the family at 1, 8 and 27. The body-centred lattice sits inside the primitive one at index two, and scaling it gives the family at 2 and 16. The face-centred lattice sits inside at index four, giving 4 and 32.

A cube, twice a cube, four times a cube, and nothing else at all. The three that stay cubic establishes exactly that, by asking which sublattices keep a symmetry rather than by enumerating centrings — and the three that come out are the three cubic Bravais lattices, the same three the fourteen produces from the other direction.

What is new here is reading it as an arithmetic of copies. Those seven indices are the entire row for any cubic space group, and there are no others at any index whatever.

The order against the room #

Setting the cubic answer beside the plane’s makes the trend a measurement rather than an impression.

Order one and two: seven hundred and sixty-two. Order two with a mirror: a hundred and fifty-six, or ninety-eight on a centred lattice. Order three: eighteen. Order four: a hundred and fifty-six for the rectangular groups, ninety-eight for cmm, twenty-four for p4. Order six: eighteen for p6 and eight for p3m1 and p31m. Order eight and twelve: eight. Order forty-eight, in space: six, and to a larger index than any of them.

Every operation of a point group is a condition a sublattice must satisfy, and the conditions are close to independent, so each one cuts the count by roughly a constant factor. The relationship is a trend rather than a law — cm and pm have the same order and different counts, because a mirror along a diagonal is a different condition from a mirror along an edge — but the direction never reverses.

The reason it never reverses is not subtle. A sublattice invariant under a group is invariant under each of its operations, so the invariant sublattices of a larger group are a subset of those of any subgroup. Adding an operation can only remove lattices, and the counts are nested exactly as the point groups are.

The rows make the same point in a different shape. p1’s has an entry at every index; p4’s is nought wherever a prime three more than a multiple of four divides it to an odd power; p6m’s has entries at eight of the first thirty indices. Reading down the figure is reading the conditions accumulate.

The rotation groups lose indices and the mirror groups lose multiplicity. p4’s row is sparse and its non-zero entries are often two or three, because a split prime gives two ideals; p4m’s row is sparser still and its entries are ones, because a mirror cannot keep one factor of a split prime without the other. The two kinds of operation thin a row in different ways, and the cubic group does both at once.

Why the cubic case collapses rather than thinning #

The plane’s rows thin and the cubic row collapses, and the difference between thinning and collapsing is worth naming.

A plane group with a four-fold rotation has invariant lattices that are the ideals of ℤ[i], and there are as many of those as there are Gaussian integers of each norm — a thin set, but an infinite family with a rich multiplicative structure. The rotation constrains the lattice into a ring and the ring still has room in it.

A cubic group does not leave a ring. Its rotations mix all three coordinate directions, so no factorisation of the kind an axial group has — an ideal across, a multiple along — is available: there is no “across” and no “along”. What is left is the sublattices invariant under the whole of a group acting irreducibly on three dimensions, and an irreducible action leaves very little.

The arithmetic that survives is scaling and two centrings. Scaling is available to every lattice whatever its symmetry, so it is not a fact about cubes; the two centrings are the only genuinely cubic content of the row, and they are two objects rather than an infinite family. That is the collapse: the row is a scaling family times a set of size three, where p4’s is a scaling family times the ideals of a ring.

So the honest statement about the cubic groups is that their copies of themselves are almost all trivial. A copy at index $m^3$ is the group drawn at m times the scale, which is a relabelling rather than a discovery; the copies at $2m^3$ and $4m^3$ are the two that say something, and they say what centring says.

What a copy at index eight is, in a crystal #

A row this short is worth reading as physics for a moment, because the two entries that are not scalings are objects a crystallographer already has a word for.

A copy of a cubic space group at index $2m^3$ is the group on the body-centred lattice scaled by m; at $4m^3$ it is the group on the face-centred lattice. In the language of structures those are superstructures: the parent describes a crystal and the copy describes an ordered version of it in a cell twice or four times as large. Which reflections a superlattice adds is the measurement that sees them — exactly n − 1 new reflections per parent cell, with intensities that are differences rather than sums.

So the cubic row says something short and definite about what a cubic crystal can order into while staying cubic: it can double its cell body-centred, quadruple it face-centred, or scale it whole, and nothing else keeps the full symmetry. Every other ordering lowers it, which is why the ordered alloys of cubic metals are so often not cubic and why the ones that are, are the ones with the two centrings in their history.

The plane’s richer rows say the corresponding thing about surfaces, where a square lattice can order into a superstructure at index five in two chiral ways — the two conjugate Gaussian ideals — and a surface scientist writes the result with a rotation in its name. That family has no cubic analogue at all.

The row and the series #

A row this sparse is an Euler product as much as any other, and its factors are short.

At the prime two the local factor is 1, 1, 1, 1 … — one lattice at every power of two, since 2, 4, 8, 16, 32 are 2·1³, 4·1³, 2³, 2·2³ and 4·2³ respectively. At the prime three it is 1, 0, 0, 1, 0, 0, 1 … — one at 1, 27, 729, and nothing between. At every other prime it is the same: one at each cube of a power and nothing else.

So the cubic row’s series is ζ(3s) times a factor at two, up to the bookkeeping, where p1’s is ζ(s)ζ(s−1). A zeta function evaluated at three times the argument grows a great deal more slowly, which is the analytic form of the same statement: the number of invariant sublattices of index at most N grows as the cube root of N for a cubic group and as $N^2$ for p1.

That contrast is the whole essay in one line, and it is worth noticing that both are zeta functions. The point group does not change what kind of object the answer is; it changes where the argument is evaluated.

One dimension at a time, and the reason three is different #

The trend across the plane groups and the collapse in space look like the same effect at two scales, and they are not quite.

In the plane, a large point group still acts on a two-dimensional lattice, and two dimensions leave room for a ring. A four-fold rotation makes the lattice a module over ℤ[i]; adding a mirror cuts the module’s ideals to the ones fixed by conjugation; and what is left is still an infinite family with a multiplicative structure, just a thinner one. The largest plane point group has order twelve and the lattice keeps a ring.

In space a cubic group acts irreducibly on three dimensions and leaves no ring at all. There is no direction to factor out, so there is no arithmetic for the constraint to thin — the invariant lattices are simply the ones a scaling and two centrings produce.

The axial space groups sit between, and their invariant lattices factor as a plane’s ring across the axis times a multiple along it. So the three cases are three different structures rather than three points on one curve: a ring, a ring times a line, and nothing. What the order of the point group predicts is the size of the answer; what predicts its shape is whether the action leaves anything to factor by.

That is the reason the trend is a trend and not a law, and it is the sharpest thing the cubic case has to say about the plane’s.

Where the exactness stops #

Computed here. For every index to forty, every sublattice of a cubic lattice in Hermite normal form, and which of them all forty-eight operations of the cubic point group carry to itself by the integrality test on H⁻¹MH; the shape of each index that carries one; and the plane’s counts to index thirty for comparison. Two checks that can fail: every index carrying an invariant sublattice must be a cube, twice a cube or four times a cube, and there must be exactly one at each.

This is a count of lattices, not of subgroups. A lattice is not a subgroup, and how many copies of a cubic space group sit on each of these seven lattices is a cohomology computation this page does not make. It is likely to be larger than one, since a cubic point group is large and has a great deal to twist by, so the sparseness of the row understates how many subgroups there are.

And it is the full cubic group. The twenty-three cubic point groups of lower order — the tetrahedral ones, the rotation groups without the inversion — have more invariant lattices than the holohedry does, since they impose fewer conditions. How many more is the same computation with a smaller operation list and is not run here; what is certain is that it is nested between the holohedry’s seven and the plane’s counts in the sense above.

Forty is where the enumeration stops, and it is the sublattice count that stops it. Every sublattice of index n in three dimensions has to be produced before it can be tested, and there are hundreds at the larger indices; the test itself is cheap. Nothing suggests an eighth invariant lattice is waiting just past forty — the next entries the three families predict are 64, 125 and 128 — but nothing here rules one out either.

The series is written and not derived. That the local factors are what they are follows from the shape of the indices, and no analytic statement about the resulting function is computed.

The refusal is a cubic sublattice of index three. Thirteen sublattices of index three exist in space; not one of them keeps the cubic symmetry, because three is not a cube and is neither twice nor four times one. A computation that found one would have a bug in the integrality test, and a computation that found none at every index would have a bug the other way — which is why the seven that do exist are as much a test as the ones that do not.

Who noticed, and in what language #

That the cubic sublattices of a cubic lattice are the primitive, body-centred and face-centred ones at their scalings is old and is usually stated as a fact about Bravais lattices rather than about subgroups. Auguste Bravais had the fourteen lattices in 1850, and which of them is a sublattice of which is elementary once they are in hand.

The subgroup reading is Billiet and Bertaut’s, from the tabulation that produced the International Tables’ subgroup volume. The cubic entries in that volume are short, and the shortness is what this page is about: a series of index $p^3$ for every prime, a few entries at powers of two, and no infinite family indexed by primes in any other pattern. A reader coming from the tetragonal or hexagonal tables, where the entries run on for a column, will find the cubic ones surprisingly bare.

The general principle — more symmetry, fewer invariant objects — has no single author because it is not a theorem. It is what a constraint does, and its precise form depends on which constraint. What makes it worth measuring rather than announcing is that the rate is not obvious in advance: an order of forty-eight might have left one lattice or a hundred, and the answer at index forty is six.

Still open: where the two effects cancel #

Two things run in opposite directions as the point group grows, and nothing here says which wins.

The lattices get fewer, which is this page. And the copies on each lattice get more, which is the census of copies: p4m has two invariant lattices to index six and eight copies on them, a factor of four, where p1 has thirty-two and thirty-two, a factor of one. A larger point group has more cocycles to twist by as well as more conditions to satisfy.

So the number of subgroups a group has at index at most N is the product of a falling count and a rising multiplicity, and whether it falls, rises or stays put is not something either computation answers on its own. The measurement that would settle it is the copy count taken to the indices where the lattice count has thinned — index thirty rather than index six — which the enumeration on that page is a factor of the index short of reaching.

The plane already shows both effects in one table. p1 has thirty-two lattices and a ratio of one; p4m has two lattices and a ratio of four; p6m has two lattices and a ratio of one. Over that narrow range the falling count wins easily, and the total number of subgroups falls with the order. Whether it goes on winning at larger indices is the open question, because the ratio is a cohomology and cohomology groups do not grow the way lattice counts shrink.

For the cubic groups the same question is sharper still, because the lattice count is as low as it goes and the point group is as large. Seven lattices and forty-eight operations could produce very few subgroups or a great many, and which it is decides whether the cubic entries in the Tables are short because there is little there or short because a great deal has been collected onto a few lines.