Two ways down from a group

Take p4m — the group of a square tile with its diagonals and its edge bisectors all mirrors — and remove half of its symmetry. There are two entirely different ways to do it, and the difference is not a matter of degree.

The first way keeps every translation and throws away operations. Delete the mirrors and what is left is p4: the same lattice, the same cell, four-fold centres in the same places, and at every one of those places a symmetry that used to be there and is not.

The second way keeps every operation and throws away translations. Every mirror and every rotation survives; what changes is that the pattern now repeats over a cell twice as large. Locally nothing has gone. Globally the repeat has doubled.

Crystallography names these in German, because Carl Hermann did: translationengleiche for the first — equal in translations — and klassengleiche for the second, equal in class. The words are forbidding and the distinction is not.

The index is the same number in both cases #

Both kinds are subgroups of index two, and the index arithmetic is indifferent to which kind is meant: the subgroup has half the operations, its fundamental domain has twice the area, and the parent’s operations fall into two cosets of it.

That indifference is the reason the distinction has to be made deliberately. A count of subgroups by index says nothing about what a pattern actually lost, and the two losses look nothing alike. A translationengleiche descent is what happens when a crystal distorts and a mirror stops being a symmetry; a klassengleiche descent is what happens when it orders and two sites that were equivalent stop being so. Those are different physical events with the same index.

Seventy-four, split twenty-nine to forty-five #

The subgroups of index two are already on this site under another name. A two-colouring of a pattern is a homomorphism onto the two-element group; its kernel is the colour-preserving half; and the enumeration of the seventeen plane groups’ two-colourings came to seventy-four.

That count can be sorted. A homomorphism either kills both lattice translations or it does not, and there is no third option: if it kills them the kernel contains the whole lattice and the subgroup is translationengleiche, and if it does not the kernel meets the lattice in a sublattice of index two, every point-group element still occurs somewhere in it, and the subgroup is klassengleiche.

Forty-five to twenty-nine is not a near-even split, and the direction of the imbalance is worth a sentence. A group’s point group is small — twelve elements at most — so the ways of deleting operations run out quickly. Its lattice is infinite, and thinning it is a question about sublattices, of which there are always more than one at every index. The second kind is more numerous for the same reason there are more lattices than point groups.

The test this site has been using could only see half of them #

The obvious containment test compares two groups’ operations modulo one shared lattice and reports whether one set contains the other. It is what the subgroup relations among the seventeen were drawn from, and it is correct.

It is also, by construction, blind to every klassengleiche subgroup. Those subgroups have a different lattice; written in the parent’s basis their translations are not unit vectors, and no comparison that reduces everything modulo the parent’s cell can see them at all. Forty-five of the seventy-four are invisible to it — not misreported, not approximated, simply outside what it is asking.

Nothing was ever wrong. The essays built on it say containment among the seventeen, which is what it measures. But a reader could reasonably have taken the resulting diagram for the subgroup structure of the plane groups, and it is a little under half of one.

Finding them without a table #

The enumeration runs inside a finite group, and the reason it can is worth stating because it is the same reason the two-colour count was finite.

A homomorphism onto a cyclic group of prime order p kills every p-th power, so it kills p times every translation. The whole question therefore lives in the quotient of the plane group by pΛ — the group modulo p-fold translations — which has p² times the order of the point group and is at most 108 elements for the primes this site asks about. Inside that finite group, every candidate homomorphism is an assignment of values to a generating set, propagated by closure, kept if it never gives one element two values.

That is the entire method. There is no list of subgroups anywhere, no formula for how many there should be, and the counts above are measurements of a search.

One practical detail turned out to matter. The obvious generating set — every coset representative of the group, plus the two lattice translations — is fourteen elements for p6m, and trying all p¹⁴ assignments takes seconds at p = 3. A minimal generating set found greedily is three or four elements, every homomorphism restricts to one on the short list, and the search finishes in milliseconds. The answer is identical; the arithmetic is the same arithmetic. What changed is only how much of it is redundant.

Naming one is harder than finding one #

A subgroup arrives as a set of operations. Saying which of the seventeen it is takes more work, and the work is instructive.

A subgroup rarely sits in the setting its standard symbol is written in. p4m’s mirror subgroup with the mirrors along the diagonals is cmm — but cmm’s standard generators are written on a rhombic basis, and comparing operation sets directly reports no match. The comparison has to run over changes of basis: every integer matrix of determinant ±1 with small entries, conjugating the subgroup’s operations into each candidate setting.

Even that is not enough. A subgroup sits where its parent put it, and a group of half-turns about (¼, ¼) is the same group as one about the origin. Moving the origin by s changes an operation’s translation part by (I − M)s, so the search runs over origin shifts too — on a grid of twelfths, because a translation part in the seventeen has denominator two or three and (I − M)s can halve or third a shift. Left at quarters, every subgroup of the hexagonal groups comes back unnamed.

At index three the answer changes shape #

Two is not representative. At index three the same enumeration finds twenty-six subgroups across the seventeen, and they are distributed completely differently: only seven of the plane groups have any at all.

The reason is a fact about the point groups rather than about the lattices. A homomorphism onto a cyclic group of order three must kill every element of order two, since two and three are coprime — so any group containing a half-turn, a mirror composed with itself, or a four-fold has already had most of its operations killed, and what remains rarely reaches onto three elements. The groups that do contribute are p1, pm, pg, cm, p3, p31m and p6.

That last entry deserves its own sentence. p3m1 and p31m are the pair this site opened with — two groups with the same operations arranged differently, which no amount of looking distinguishes. They are also a group and a subgroup, of index three, of the klassengleiche kind, on a cell three times as large turned by thirty degrees. That relation is invisible to a same-lattice containment test, which is why every comparison of the pair by its operations misses it — and it is why neither group has a copy of itself at index three.

A klassengleiche subgroup is a choice of sublattice together with a choice of which coset representatives to keep, and the first half of that is pure lattice arithmetic.

That bound is the useful half of the split. Counting klassengleiche subgroups of index n means counting sublattices of index n that the point group preserves, and then counting the extensions on each — which is a question with two independent halves, and the first half is a piece of arithmetic with no crystallography in it.

The third name, which is a special case of the second #

There is a subdivision of the klassengleiche half that the counts above do not separate and that the literature always does, because it behaves differently.

A klassengleiche subgroup keeps the point group and thins the lattice, and sometimes the result is a group of the same type as the one it came from — p1 inside p1 with a doubled cell, p4 inside p4 with a cell four times as large. Those are called isomorphic subgroups, and there are infinitely many of them for every group, one or more for each index at which a suitable sublattice exists.

The distinction earns its name because of what it does to a census. A count of subgroups of index two is finite, and so is a count at index three, but a count over all indices is not — the isomorphic subgroups run away to infinity in a completely regular manner, and any table of subgroups has to either bound the index or describe the isomorphic ones by a rule rather than listing them. The International Tables do exactly that: maximal translationengleiche and non-isomorphic klassengleiche subgroups are listed, and the isomorphic ones are given as a series with a parameter.

So the two ways down are really two ways down and one of them has a tail. The tail is the least interesting part structurally and the most common in practice, because a superstructure produced by ordering on a sublattice is very often isomorphic to its parent, and what changed is the cell rather than the symmetry.

What the distinction predicts about a transition #

The split is not vocabulary, and the sharpest evidence for that is a rule about phase transitions that uses it directly.

A crystal undergoing a continuous transition passes from a high-symmetry phase to a low-symmetry one, and the low-symmetry group is a subgroup of the high-symmetry group. Landau’s theory adds two requirements: the subgroup must be maximal — nothing sits properly between them — and the quantity that becomes non-zero at the transition must transform as a single irreducible representation of the parent group.

The two ways down then produce two different sets of consequences, and both are measurable.

A translationengleiche descent loses operations and keeps the cell. The lattice is unchanged, so no new reflections appear; what appears is a splitting of reflections that were equivalent and are no longer, and a set of orientational domains — regions that chose different survivors of the lost operations, meeting at twin boundaries.

A klassengleiche descent keeps the operations and loses translations. The cell grows, so new reflections appear at positions the parent lattice does not use — superlattice peaks — and the domains are translational rather than orientational: regions that ordered on the same sublattice with different offsets, meeting at antiphase boundaries.

New spots or split spots; antiphase boundaries or twin boundaries. Those are different observations, they are made with the same instrument, and which one occurs is fixed by which half of the subgroup census the descent came from — decided before the material is cooled.

Why anybody cares, in two subjects that do not cite each other #

The distinction is not bookkeeping. It decides what a physical descent can do.

When a crystal distorts at a phase transition and loses a mirror, the descent is translationengleiche: the cell is unchanged, and the domain states are orientations of the same structure. When a crystal orders — when two atom types that were sharing a site sort themselves out — the descent is klassengleiche: the point group is unchanged and the cell doubles, and the domain states are antiphase domains, identical in orientation and out of step by a translation the parent had and the child does not.

Those are the two things a solid can do to its own symmetry, they produce different defects, they show different diffraction, and the group theory tells them apart before any of that is measured. The tree of descents that structural crystallography draws — Bärnighausen’s — labels every edge with t or k and its index, for exactly this reason.

Who separated them, and when #

Carl Hermann introduced both words in 1929, in a paper on the subgroups of the space groups, and the terms have been used untranslated ever since — partly because the English versions are clumsy and partly because the German ones are exactly as long as they need to be.

What he was after was not a taxonomy. The question of the period was how one space group sits inside another, because a crystal that changes structure changes group, and the relation between the two groups constrains what the change can do. Hermann’s theorem is the one that makes the distinction load-bearing: every maximal subgroup of a space group is either translationengleiche or klassengleiche, never a mixture. A subgroup that loses some operations and some translations is always reachable as one descent of each kind, one after the other — so the two words are not merely a convenient sorting, they are a decomposition.

The theorem is why the trees in structural crystallography have labelled edges rather than annotations. Hartmut Bärnighausen set them out in that form in 1980, and the label on each edge is a letter and an index — t2, k3 — which between them say everything a structural relation says. Every path down such a tree is a sequence of one-step descents, and the theorem guarantees that no relation is missed by insisting on single steps.

Where the exactness stops #

Everything above is a statement about groups of index two and three with cyclic quotient. That is not every subgroup of index two — index two is always cyclic, so at p = 2 the enumeration is complete — but at higher composite indices there are subgroups whose quotient is not cyclic, and this method does not find them. The count of twenty-six at index three is complete for the same reason two is: three is prime.

What is not attempted here at all is the maximal subgroup relation in full — the graph of every subgroup of every index, which is infinite in both directions, since every group contains sublattice copies of itself at every index. What is finite and worth having is the list of maximal subgroups, and reaching that needs the same enumeration run at every prime index with a proof that nothing above some bound is maximal. The bound exists and is not derived here.

And the sorting itself rests on a convention that is worth naming: index is defined against the parent, so a subgroup of a subgroup is counted twice over in these lists, once at each level. The seventy-four are subgroups of the seventeen, not of one another.

Where the ladder goes next #

Two directions, and both are open.

The lattice half — how many sublattices of index n there are, which of them keep a symmetry, and what arithmetic decides it — turns out to be a question about sums of two squares, and it is the next rung of a different ladder rather than of this one.

The group half is the maximal-subgroup graph itself: which of these seventy-four cannot be reached by composing two smaller descents, and what the resulting tree looks like when the klassengleiche edges are drawn as well as the translationengleiche ones. That tree is the one structural crystallography actually uses, and this site has drawn a little under half of it for five phases.

The two lists are also why the distinction had to be made deliberately rather than emerging from a count. Index two is index two either way, the arithmetic of domains is the same arithmetic, and nothing about the number says which kind produced it. What differs is everything downstream — the reflections, the boundaries, the defects — and none of that is visible in the subgroup census itself.