The average that makes it finite

Two sentences appear in this collection more often than almost any others, and neither has ever been proved here.

“A finite group of isometries fixes a point.” It is why the plane’s finite groups are the cyclic and dihedral ones, and why a crystal’s point group can be drawn as a stereogram about a centre.

“A finite group of integer matrices preserves a positive definite form, so it is the symmetry group of some lattice metric, so its order is at most twelve in the plane.” It is why the thirteen arithmetic classes can be found by searching inside two holohedries, and why there is a list at all.

They are the same argument, used on different objects, and the argument is: average over the group.

Average the points #

Take any point and let the group act on it. The images form an orbit, and if the group is finite the orbit is finite, so it has a centre of mass.

Now let any element g of the group act on that centre. Averaging is linear and g permutes the orbit, so g sends the mean of the orbit to the mean of the same set of points listed in a different order — which is the same point. Every element fixes the centre of mass.

That is the whole of it. A plane isometry with a fixed point is a rotation or a reflection about that point, so a finite plane group is a group of rotations and reflections about one centre; the rotations form a cyclic group and one reflection doubles it, so the group is a Cₙ or a Dₙ. The classification that this collection uses to draw every point group follows from a mean.

The finiteness is doing all the work. The right-hand panel above shows a point translated repeatedly. Every partial average exists and none of them is the answer — the mean of the first ten is not the mean of the first hundred — because the orbit never closes. A group with a translation in it has no fixed point, and that is the difference between the thirty-two crystal classes and the two hundred and thirty space groups.

It is worth noticing how little the argument uses, because the smallness is the reason it turns up in three different places later in this essay. It never mentions the plane; it never mentions how many elements the group has, only that the number is finite; and it never mentions what kind of object is being averaged, only that averaging makes sense and that the group acts on the objects by permuting them. Every one of those is a hole through which the same three lines can be pushed at something else. Average a point over an orbit and a finite group of isometries acquires a fixed point. Average a quadratic form and a finite group of matrices acquires an invariant metric. Average a lattice and a finite group of integer matrices acquires an invariant lattice. The three arguments below are the same argument with a different object in it, and the only thing that ever has to be checked is that the object can be averaged at all.

What the fixed point is worth #

The centroid argument is three lines long and it decides the shape of two whole classifications, so it is worth being precise about what it delivers and what it does not.

It delivers a common fixed point, not a fixed point for each element. Any single rotation has a fixed point; the content is that all of them share one. That is what makes a stereogram possible: every operation of a crystal class can be drawn about the same centre, and a class is a picture rather than a list.

It delivers it for any orbit. The centre of mass of the orbit of any starting point is fixed, so there is no need to choose a good one — and if two starting points give different centres, both are fixed, so the whole line between them is fixed and the group is a group of reflections in that line or the identity. The construction is insensitive to its own input, which is the mark of a real argument rather than a lucky one.

It says nothing about which finite groups occur. That a finite group fixes a point reduces the classification to subgroups of the orthogonal group; it does not enumerate them. The enumeration is the axis equation, and it is a separate piece of counting.

Average the metric #

The second use is less familiar and more powerful.

Let G be a finite group of invertible matrices — not necessarily orthogonal, not necessarily nice. Put

$$A = \frac{1}{|G|}\sum_{M \in G} M^{\mathsf{T}}M .$$

Each summand is symmetric and positive semi-definite, and the identity’s summand is the identity matrix, so A is symmetric and positive definite. And for any N in G,

$$N^{\mathsf{T}}AN = \frac{1}{|G|}\sum_{M} (MN)^{\mathsf{T}}(MN) = A,$$

because M ↦ MN permutes G. So A is an inner product that every element of G preserves: G is a group of isometries of some metric, even if it was not a group of isometries of the standard one.

The consequence is a bound. A group preserving a positive definite form is conjugate to a group of orthogonal matrices — change basis to make A the identity — so all its elements have bounded entries, so a group of integer matrices preserving one is finite. Run it the other way and it says that a finite group of integer matrices is the symmetry group of some lattice metric, which is a lattice’s holohedry or a subgroup of one, so its order is at most 12 in the plane and at most 48 in space.

That is the sentence the arithmetic-class enumeration takes for granted, and it is now a computation: the averaged form is built in exact rational arithmetic and MᵀAM = A is checked for every element.

The space of invariant forms #

Averaging produces an invariant form. Solving the invariance equations directly produces all of them, and the difference is informative.

Writing A as [[a, b], [b, c]] gives three unknowns; each generator contributes three linear equations; the solution space is computed by elimination over the rationals. For each of the skewed groups here the space comes out one-dimensional — so up to scale the averaged form is the only invariant metric, and the group determines its own geometry completely.

The exception in that table is the group of order two. Conjugating −1 by anything gives −1 back, so it cannot be skewed at all, and every symmetric form is invariant under it: the solution space is all three dimensions. A group whose action is irreducible pins its metric down to a scale factor; a group acting by a scalar pins down nothing. The dimension of the space of invariant forms is a measure of how much of the geometry the group decides, and for the groups that matter here it decides all of it.

What “conjugate to orthogonal” means in practice #

The step from “there is an invariant positive definite form” to “the group is conjugate to a group of orthogonal matrices” is a change of coordinates, and it is worth seeing as a construction rather than as an existence claim.

A positive definite form A factors as A = BᵀB — by Cholesky, or by taking any square root — and then for M in G,

$$M^{\mathsf{T}}AM = A \iff (BMB^{-1})^{\mathsf{T}}(BMB^{-1}) = I,$$

so BMB⁻¹ is orthogonal. The group has been rotated into the orthogonal ones by a single change of basis, and the basis is computed from the form the group itself produced.

The consequence used everywhere here is that the entries are then bounded: an orthogonal matrix has entries in [−1, 1] and there are finitely many integer matrices with bounded entries, so a group of integer matrices preserving a positive definite form is finite. Running the implication in that direction is the proof that the arithmetic classes are a finite list, and running it in the other is the proof that each of them sits inside a holohedry.

B is not integral and does not need to be. The change of basis that orthogonalises the form is a real matrix; the change of basis that integralises the group is a rational one, computed differently in the next section. The two are separate constructions doing separate jobs, and conflating them is the easiest mistake available here.

Average the lattice #

The third use turns rational matrices into integer ones.

Suppose G is a finite group of matrices with rational entries. Take the two standard basis vectors and let G act; the ℤ-span of the resulting orbit is finitely generated, is full rank because it contains the basis, and is carried onto itself by every element of G because acting permutes the generators. So it is a lattice that G preserves, and in a basis of that lattice every element of G is an integer matrix.

This is why “finite subgroup of GL(2,ℚ)” and “finite subgroup of GL(2,ℤ)” are the same classification, and it is the step that makes the thirteen arithmetic classes a classification of anything rather than of one particular coordinate system.

The lattice is not unique and nothing here pretends otherwise. A different starting orbit gives a different invariant lattice — the one computed here contains the integers with index four, and there are others. What is unique is the conclusion: some invariant lattice exists, and one is enough.

What the round trip checked, and how #

A shear must not close. The matrix with ones on the diagonal and a one above it has infinite order, and the closure has to run away rather than return a group.

A shear must have no invariant metric. Solving directly, the space of symmetric forms a shear preserves is one-dimensional and every form in it has determinant zero: it measures nothing along the shear direction. That is not the averaging trick failing, it is the trick detecting what it is for — a group with an invariant positive definite form is bounded, so a group with no such form is unbounded, so it is infinite.

A set that is not a group must not average to something invariant. This refusal needed care, and the first version of it was worthless. Averaging over any set of orthogonal matrices returns the identity form, which every orthogonal matrix preserves — so a subset of a rotation group passes the test while proving nothing. The test only bites on matrices that are not orthogonal, which is why the example is drawn from a skewed group, and there the average of two of the three elements is invariant under neither.

An infinite orbit must have no centre of mass. The mean of the first n images under a translation is computed for three values of n and has to be seen moving.

Where the exactness stops #

Averaging needs the group order to be invertible. Dividing by |G| is fine over the rationals and over the reals and is exactly what fails in modular representation theory, where the characteristic divides the group order and the averaging argument — Maschke’s theorem — stops working. Nothing in this collection is over a finite field, so the trick is always available here; that it is not always available is worth knowing.

Positive definiteness comes from the identity being in the group. Each summand MᵀM is only positive semi-definite, and the sum is definite because one of the summands is the identity. Averaging over a set that omits the identity can produce a degenerate form even when the set is otherwise well behaved.

The invariant lattice’s index is an artefact of the starting orbit. The construction here reports index four over the integers, and a reader could reasonably take that number for a property of the group. It is not: it is a property of the orbit that was used, and starting from a different vector gives a different index. What is a property of the group is that some invariant lattice exists, and every statement made here rests only on that.

None of this gives a bound on the order by itself. It reduces “how large can a finite group of integer matrices be” to “how large can a lattice’s symmetry group be”, which is then answered separately — by the enumeration in the plane, and by Minkowski’s bound in general. The averaging is the bridge, not the answer.

And the bridge is one-way in a sense worth stating. Averaging turns a finite group into a group of isometries; it does not turn a group of isometries into a finite one. An infinite group can preserve a metric perfectly well — the full rotation group of the plane does — so nothing here says that preserving a form is enough for finiteness. What makes the implication run is the integrality: a group of matrices that both preserves a positive definite form and has integer entries is finite, because the first condition bounds the entries and the second makes the bounded region contain finitely many points. Drop either half and the conclusion goes. That is why the averaging argument appears in this collection attached to lattices rather than to groups in general, and why the number it eventually produces — thirteen arithmetic classes in the plane, and a bound rather than a list above three dimensions — is a fact about integer matrices and not about symmetry.

The same trick, three times, in one sentence #

The three sections above are the same computation applied to three different kinds of object, and the pattern is worth stating on its own because it recurs far outside this subject.

Given a finite group G acting on anything with an average, average an arbitrary element of that thing over G and the result is G-invariant. Points average to a fixed point. Inner products average to an invariant inner product. Lattices — where the “average” is the span of an orbit rather than a mean, because lattices add rather than divide — give an invariant lattice. Probability distributions average to invariant measures; functions average to symmetric functions; in representation theory the same step is Maschke’s theorem and it is what makes every representation of a finite group a sum of irreducible ones.

The character sums this collection uses to decide which tensor components a crystal class permits are the same averaging again: projecting a general tensor onto the part the group leaves alone is exactly averaging the tensor over the group. Neumann’s principle, in the form the machinery here computes it, is this essay’s trick applied to a physical property.

That is one reason to give the step its own essay rather than a footnote. It is not a lemma used twice; it is the reason half the machinery on this site works.

Why the shear is the right counterexample #

Every refusal here is the shear, in one disguise or another, and that is not laziness — it is the smallest thing that breaks each argument, which is what a counterexample should be.

It has determinant one, so it preserves area; it is an integer matrix, so it lives in the same GL(2,ℤ) the arithmetic classes are drawn from; it fixes a line pointwise, so it has plenty of fixed points. Everything about it looks like a symmetry of a lattice, and it is one — the integer lattice is carried onto itself by it exactly. What it is not is finite, and the only symptom is that its powers never come back.

So a lattice does have infinitely many symmetries in the sense of area-preserving integer maps, and the classification here is of the ones that preserve a metric as well. The shear is the standing reminder that the metric is what makes the list short, and that dropping it does not give a longer list but no list at all.

Where the ladder goes next #

With a fixed point and an invariant metric in hand, the classification of finite groups is a classification of subgroups of the orthogonal group, which is where the five families and the five solids come from. The remaining question is what the lattice then adds — and the answer is the crystallographic restriction, which this collection reaches from four different directions and which is, in the end, one more statement about integers being integers.

The one hypothesis, and what happens without it #

Every use of the trick divides by the order of the group. That looks like bookkeeping and it is the hypothesis — the place the argument fails, and it fails for a reason worth knowing.

Averaging needs 1/|G| to exist. Over the rationals or the reals it always does, which is why every use above goes through. Over a field of characteristic p, it does not when p divides |G|: the sum of |G| copies of anything is zero, and the average is undefined.

That single failure is the whole of the difference between ordinary and modular representation theory. With the average available, every representation of a finite group breaks into irreducible pieces — that is Maschke’s theorem, and the proof is this trick applied to a projection. Without it, a representation can have a subrepresentation with no complement, so it cannot be decomposed, and the tidy picture this collection relies on everywhere collapses.

Nothing in crystallography meets that case, because the matrices here have rational entries and the fields are the rationals and the reals. But it is worth knowing which hypothesis is load-bearing: not finiteness alone, and not the group being of matrices — the divisibility. A statement in this collection that “the group is finite, so average” is using two facts, and only one of them is about the group.

The same average, three essays over #

The construction has appeared in this collection under another name and it is worth connecting them, because a reader who has met one will not have recognised the other.

Neumann’s principle counts a crystal’s independent property components by averaging the group’s action on tensors and taking the trace. That average is this average, with the group acting on a space of tensors instead of on a point or on a form, and the fact that it produces something invariant is the same three-line argument.

So the collection’s property counts, its fixed-point argument, its metric argument and its lattice argument are four applications of one operator. Each of them takes an arbitrary object, averages it over the group, and gets an invariant one — and each is exact because the group is finite and the coefficients are rational.

That is why the four never disagree, and it is a reasonable summary of what a finite group is good for: it is small enough to sum over, and summing over it is how everything invariant gets made.