Which groups a crystal could have

What forces a lattice proves Bieberbach’s first theorem, and every hypothesis in it is about a group acting. The group is discrete, meaning its elements do not accumulate; no point of the space is far from the orbit of any other; the elements are motions, meaning they preserve distance. Those are conditions on a group together with a space and an action, and none of them is a property the group has on its own.

That is not a defect of the theorem. It is the reason the theorem is about crystals. But it leaves a question the theorem cannot answer: hand over an abstract group, with no space attached — a presentation, a multiplication table, a set of matrices — and ask whether it is the symmetry group of some pattern in some dimension. The hypotheses have nothing to test.

Zassenhaus answered it, and the answer is three clauses. A group is a crystallographic group of some dimension exactly when it has a normal subgroup that is

1. free abelian of finite rank,
2. of finite index, and
3. maximal among its abelian subgroups.

The rank is then the dimension, the subgroup is the translations, and the quotient is the point group. Nothing in that mentions a space, a distance or an orbit.

The seventeen carry their own certificates #

The easy direction is to check that the seventeen satisfy it, and the check is worth doing group by group rather than in general, because three of the clauses are three different kinds of statement.

Free abelian of finite rank is the lattice. A plane group’s translations are the integer combinations of two independent vectors, so they are a copy of the integers in two directions, and nothing in them has finite order except the identity.

Finite index is the point group. Two operations differ by a translation exactly when their linear parts agree, so the cosets of the translations are in one-to-one correspondence with the distinct linear parts — and the operations recorded for a group here are already the operations modulo translations, so the index is the length of that list.

Maximal abelian is the clause that does real work, and it is the one that needs a calculation rather than a definition.

The indices come out 1, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 6, 6, 6, 8, 8 and 12, which is the list of orders of the seventeen point groups, arrived at from the operation lists rather than read from a table. The rank is two in every row, because there is nowhere else for it to be. And the last column is one in every row.

Nothing outside the translations commutes with all of them #

The third clause looks like seventeen separate facts and is one.

Compose an operation (M, t) with a translation (I, v) one way round and the translation comes out rotated: (M, Mv + t). Compose them the other way and it does not: (M, t + v). Those agree for every lattice vector v exactly when Mv = v for every one of them, and a linear map of the plane fixing two independent vectors is the identity. So the only operations commuting with the whole lattice are the translations themselves, in every plane group whatever, for a reason that mentions no particular group.

That is the same calculation the centre of a plane group comes from, asked about a different subgroup. There the question was which elements commute with everything, and the answer was the lattice vectors the point group fixes — nothing at all for thirteen of the seventeen. Here the question is which elements commute with the translations, and the answer is the translations, always. One calculation, two questions, and the second is uniform where the first is not.

Three groups, each failing one clause #

A characterisation with three clauses is only worth its length if each one turns something away. Three groups do the turning, and each of them is a group anybody might have supposed was crystallographic.

The free group on two letters has no non-trivial abelian normal subgroup at all, so the first clause has nothing to apply to. That is a real exclusion rather than a technicality: a free group is discrete in plenty of actions and it grows exponentially, where a crystallographic group’s orbit of a point grows as a power of the radius — the power being the dimension.

ℤ² × ℤ/2 is the interesting one, because it passes the first two clauses.

Take the plane’s translations and attach to each one a label that is nought or one, adding the labels modulo two. The result has ℤ² sitting inside it as a normal subgroup of index two, free abelian of rank two. Without the third clause it would be a crystallographic group, and it is not one.

The reason no crystallographic group looks like this is four lines and is worth having, because it is a fact about the whole class. Suppose N is a finite normal subgroup. Conjugating a translation by an element of N gives a translation, and the difference lies in N because N is normal — but it also lies in the lattice, and N meets the lattice trivially since a lattice has no element of finite order but zero. So every element of N commutes with every translation, so by the calculation above its linear part is the identity, so it is a translation, so it is the identity. A crystallographic group has no finite normal subgroup other than the identity, and the third clause is what encodes that.

The nearest miss has everything but the index #

The third group is the one that decides what has to be settled next.

Write elements as triples (a, b, c) of whole numbers with the product (a, b, c)(a′, b′, c′) = (a + a′, b + b′, c + c′ + a b′). The third coordinate remembers which factor came first, which is the whole content of the group. The elements with a = 0 form a subgroup: it is abelian, since the correction term a b′ vanishes there; it is normal, since conjugating changes only the third coordinate; it is free abelian of rank two. And nothing outside it commutes with all of it, so it is maximal abelian.

Its index is infinite. There is one coset for every value of a, and that is the only clause that fails.

And it is not an artificial example. The discrete Heisenberg group acts on three-dimensional space, discretely, with a compact quotient — by affine maps rather than by motions. It is as close as a group can come to being crystallographic without being so, it fails by exactly the clause Bieberbach’s theorem has to prove rather than assume, and the reason it is not a group of motions is what the affine case settles.

What the rank knows #

The clauses do more than decide membership. They name the dimension.

The rank of the free abelian subgroup is the number of independent translations, and Bieberbach’s theorem says a crystallographic group in n dimensions has n of them. So an abstract group satisfying the three clauses can be examined and told which space it belongs to, without being shown a space at all.

Placing it in the wrong dimension fails one of the two acting hypotheses rather than the algebra, and which one depends on the direction of the error. Too few dimensions and the translations cannot be independent, so the action is not discrete. Too many and there is a direction the group never repeats in, so points arbitrarily far along it are arbitrarily far from every orbit — the covering radius is unbounded, which is the failure the frieze exhibits in the plane, measured there on a disc.

So a crystallographic group cannot be crystallographic twice. That is a stronger statement than it looks, because the analogous claim fails for other classes of group: a free group acts on trees of many valencies, a surface group on the hyperbolic plane in a continuum of ways. Here the group fixes the space.

What the clauses do not ask for #

Three short clauses invite the suspicion that something has been left out, and it is worth saying what has and has not.

Finite generation is not asked for and follows. A group with a free abelian subgroup of finite rank and finite index is generated by a basis of that subgroup together with one element from each coset, which is finitely many. So a crystallographic group is finitely generated without the clauses mentioning it, and written as generators and relations a plane group takes two to four letters.

Torsion-freeness is not asked for and does not follow. Thirteen of the seventeen contain a rotation, which has finite order. The clauses admit them, and they should: a wallpaper pattern with four-fold centres is as much a crystal as one without. The crystallographic groups with no element of finite order are a proper subclass — the ones whose quotient is a manifold rather than an orbifold — and there are two in the plane and ten in space.

And the point group is not asked to be anything. Nothing in the three clauses says the quotient must be a group of order dividing twenty-four, or must contain only rotations of order two, three, four and six. Those are consequences of the quotient acting faithfully on a lattice by integer matrices, which is the crystallographic restriction and is a theorem rather than a hypothesis. The clauses are deliberately weaker than the list of things a crystallographic group turns out to satisfy, which is what makes them a test rather than a summary.

The same clauses with rank one describe a frieze. A group whose maximal abelian normal subgroup is free of rank one and of finite index is crystallographic in dimension one, and there are two such groups up to isomorphism — the integers and the infinite dihedral group. Acting on a strip in the plane rather than on a line they are the seven friezes, which are seven because the action has more room than the group does. The clauses count groups; the friezes count actions, and the two counts differ for a reason worth keeping in view.

What this adds to the theorem below it #

The two statements are not the same one turned round, and the difference is worth naming.

Bieberbach’s first theorem is an existence result. Given the action, it produces a lattice that was not assumed. The work is entirely in producing it: the commutator that turns out to be a translation, the rotated copies that would accumulate, the covering radius that would be unbounded.

Zassenhaus’s characterisation is a recognition result. It says what to look for once the lattice is there, and it says that nothing else needs looking for. The work is in showing that the three clauses are enough — that a group with such a subgroup can be given an action, which means building a space and a representation out of the algebra rather than finding algebra inside a space.

Between them they turn a property of an action into a property of a group, and that is what makes Bieberbach’s second theorem do its work. Two crystallographic groups that are abstractly isomorphic must have the same dimension, since the rank is determined; the isomorphism must carry translations to translations, since the subgroup is determined; and what is left is an isomorphism of point groups together with a compatible map of lattices, which is an affine equivalence. The subgroup being canonical is the engine, and the third clause is what makes it canonical: without maximality there would be a choice of which abelian normal subgroup to call the translations, and an isomorphism would not have to respect a choice.

The clauses in three dimensions and above #

Nothing in the three clauses mentions two dimensions, and nothing in the calculation above does either except one step: that a linear map of the plane fixing two independent vectors is the identity. In n dimensions the same statement holds with n vectors, so the maximality argument runs unchanged, and the certificate for each of the two hundred and thirty has the same shape as the certificates above with rank three in place of rank two.

The counts of point-group orders change and nothing else does. Minkowski’s bound limits the index to a divisor of twenty-four in the plane and forty-eight in space, so the second clause is not merely satisfied but satisfied in a bounded way — and that bound, together with the finiteness of the classes of matrix group and the finiteness of the extensions, is what makes the list of groups satisfying the clauses finite in each dimension.

The characterisation itself says nothing about finiteness. It is a test one group at a time, and a test that any number of groups could pass. Finiteness is three further theorems, and keeping the two apart is the point of having a characterisation at all: it separates being a crystallographic group, which is a property, from how many there are, which is a count.

Where the exactness stops #

Computed here. For each of the seventeen: the rank of its translations, the index of that subgroup as the length of its operation list, and the number of its operations whose linear part fixes every lattice vector. For each of four abstract groups: whether the candidate subgroup is closed, abelian and normal on a sample of elements, and whether anything outside it commutes with all of it. The verdicts come from the normal forms rather than from the samples, so they do not depend on how far the sample was taken; the samples exercise the claim rather than establishing it.

Not proved here. The hard direction of the characterisation — that the three clauses are sufficient, so that a group satisfying them can be realised as a group of motions. That construction is Zassenhaus’s and is not reproduced. What is checked is the easy direction on seventeen cases and the necessity of each clause on one case each.

The samples are small and their job is small. Six elements per group, chosen to include the identity, both generators and a product or two. A claim that a subgroup is normal is exactly the kind of claim a sign error breaks, and exercising it on actual elements is what catches that.

And “crystallographic” here means what Bieberbach’s hypotheses mean. Discrete, with a compact quotient, by isometries of a Euclidean space. Every one of those three can be varied, and varying the last is the question that follows.

The refusals are the two groups that pass two clauses each, and they are refused for different reasons in the same table — one because its maximal abelian normal subgroup has torsion in it, the other because the index is infinite. A test that turned away only groups failing every clause would be a test nobody needed.

Who stated it, and why it took until 1948 #

Bieberbach proved his three theorems between 1911 and 1912, and the characterisation is Hans Zassenhaus’s, from the paper of 1948 that also gave the algorithm for enumerating the groups in any dimension. The gap is not an accident of attention. The language the clauses are stated in — normal subgroups, quotients, group extensions — was being assembled through the 1920s and 1930s, and the idea that a class of geometric objects might be characterised by a property of the abstract group belongs to that period rather than to Bieberbach’s.

What Zassenhaus wanted it for was the algorithm. An enumeration in four dimensions cannot proceed by looking at patterns, because nobody can look at a four-dimensional pattern; it proceeds by building candidate groups algebraically and testing them, and a test is exactly what the characterisation is. The four-dimensional count of 4,783 was produced that way thirty years later, and every count above it since.

Still open: the test, run on a group nobody chose #

The clauses are checked here on the seventeen, which is the case where the answer is known four other ways, and on four groups chosen because each fails one clause. That is a test of the statement and not a use of it.

The use it was designed for is the other way round: hand the test a group that arrived from somewhere else and see what it says. Two families are worth handing it and neither is handed it here. The groups generated by two or three reflections with given angles between their mirrors — Coxeter groups — are crystallographic exactly when their diagram is one of a short list, and running the three clauses on them would rederive that list from the algebra rather than from the geometry of the mirrors. And the fundamental groups of the flat manifolds, which are the crystallographic groups with no element of finite order at all, have their own characterisation on top of this one — torsion-freeness — and the ten in three dimensions are where it would be checked.

The second is the more interesting test, because torsion-freeness is a property of the group that has a geometric meaning the clauses do not mention: it says the action has no fixed points anywhere, so the quotient is a manifold rather than an orbifold. Whether that property has a statement in the same vocabulary — a clause about the extension rather than about the elements — is a question the characterisation raises and does not settle.