The descent with no shortcut

A crystal cooling through a phase transition loses symmetry, and the group it ends with sits inside the group it began with. The two ways down sorts the seventy-four subgroups of index two among the seventeen plane groups into twenty-nine that keep the lattice and forty-five that keep the point group. What that essay does not ask is whether a subgroup has to be one or the other.

It does not — in general. A subgroup can perfectly well give up some operations and some translations, and plenty do. What cannot happen is that such a subgroup is maximal.

The claim, and why it is worth having #

Hermann’s theorem. A maximal subgroup of a plane or space group is either translationengleiche — keeping every translation — or klassengleiche, keeping every linear part. Never partly one and partly the other.

The German names are worth keeping because the English ones are longer and no clearer: equal-translation and equal-class. A translationengleiche subgroup lives on the same lattice with fewer operations, which is what a crystal does when a rotation axis disappears. A klassengleiche subgroup has the same operations on a coarser lattice, which is what a crystal does when it doubles a cell, as in the ordering transitions centring as a sublattice counts.

The theorem is what makes the descent of a real crystal followable. A transition that loses operations and translations together is a composite: the theorem says there is an intermediate group, and the descent factors into steps each of one kind. So a Bärnighausen tree — the diagram a solid-state chemist draws between a parent structure and its ordered derivative — needs exactly two kinds of edge and no third, and every path is a sequence of steps whose kinds can be read off separately.

Without it, the diagram would need an edge for every mixed subgroup and would be a description rather than an argument.

There is a second reason to want the theorem, and it is the one that makes the arithmetic in this essay finite. A space group has infinitely many subgroups — every sublattice of every index gives one — so a table of them all is impossible, and a table of the maximal ones is short. Every subgroup whatever is reached by a chain of maximal steps, so the short table generates the infinite one, and the theorem is what says each step of such a chain is of a single kind.

Every maximal subgroup, by closure #

The subgroups that keep every translation are the easy half, and easy here means finitely checkable rather than short. A translationengleiche subgroup is determined by which cosets of the translation lattice survive — that is, by a subgroup of the finite group of operations modulo translations, whose order is at most twelve. So the enumeration closes every subset of one, two and three operations and keeps the distinct results, which is complete rather than sampled.

That is worth stating because this site had a partial answer already and it was not obviously partial. The machinery this site already had reaches subgroups through the quotient by pΛ, which is the right construction for the klassengleiche half and returns nothing at all at index three in p6m. The group cmm sits there, three times over, and the earlier machinery could not see it. Chains built on that edge list would have counted four descents from p6m where there are eight.

Every subgroup of prime index is maximal, and that takes no theorem: an intermediate group would have an index dividing p and greater than one, and a prime has no such divisor. So the index-two and index-three subgroups are all maximal, and testing Hermann’s statement on them tests it where it has the most to say.

The test is not the classification the code already makes. It is the two questions asked separately — are all of the parent’s translations here? and are all of the parent’s linear parts here? — with the assertion that exactly one answer is yes. A subgroup answering no twice would refute Hermann. A subgroup answering yes twice would be the parent. Over a hundred subgroups, at index two and index three, neither cell has anything in it.

The index-two split, which was already recorded #

The tally at index two comes to twenty-nine translationengleiche and forty-five klassengleiche, which is exactly what the two ways down recorded when it counted seventy-four subgroups by a different construction. Two enumerations of the same objects, one through kernels of homomorphisms onto ℤ₂ and one through closures of subsets, agreeing on a split that neither was written to reproduce.

At index three the numbers are four and twenty-two. The asymmetry is not an accident of small numbers: a klassengleiche subgroup of index p exists whenever the lattice has a sublattice of index p that the point group keeps, and the sublattice count says there are σ(p) sublattices to try at every prime. A translationengleiche subgroup of index three needs an operation of order three to give up, and only the five hexagonal and trigonal groups have one.

The chains, and why they are all the same length #

The eight routes from p6m to p1 all have three steps. That is not a feature of this example. A chain of maximal subgroups is a composition series, and the Jordan–Hölder theorem says that any two composition series of a group have the same length and the same multiset of indices — up to order. Here |p6m| = 12 = 2 × 2 × 3, and every route spends those three primes in some order.

So a crystal descending from a p6m parent to a p1 daughter through a sequence of transitions has eight itineraries available and cannot take a shorter one. The intermediate groups differ — p6, p3m1, p31m and cmm are all reachable in one step, and they are genuinely different symmetries with different domain counts — but the number of transitions is fixed by arithmetic before any chemistry is considered.

That connects directly to how many domains a transition makes, where the number of domain states is the index of the child in the parent. A three-step descent with indices 2, 2 and 3 produces two, then two, then three choices; a crystal that went down all three steps has up to twelve domain states, which is the total index. The product is invariant even though the order is not.

What a maximal step looks like in the pattern #

The arithmetic above is about operations, and the whole of it can be seen in a drawing. A subgroup of index two needs twice as much of the cell to rebuild the pattern from, because it has half as many operations to rebuild it with — which is the index arithmetic that turns a containment into a picture.

Reading that figure as a transition: a crystal in the p6m arrangement cools, its mirrors go, and the piece of the pattern that has to be specified independently doubles. Nothing has moved. What has changed is how much of the cell is free, and the freed half is where the two domain states of the transition differ from one another.

The same figure drawn for a klassengleiche step looks entirely different, and that is the visible content of Hermann’s theorem. A klassengleiche subgroup keeps every operation and doubles the cell: the fundamental domain has the same shape and the same operations acting on it, and there are simply two of them per new cell where there was one. So the two kinds of descent are distinguishable by looking, not only by arithmetic — one takes ink out of the domain and the other repeats the domain. A subgroup that did both at once would be a step in which the domain grew and changed shape, and the theorem says that a step of that kind can always be cut in two.

The same index arithmetic runs in the square system unchanged: p4 sits inside p4m at index two, so a fundamental domain for p4 is twice the area of one for p4m, and every maximal step in the tree above is a picture of that kind. The index is a ratio of areas rather than a label on an edge, and it is worth reading it that way at least once, because it is the reading a transition makes physical — the freed area is exactly what the two domain states have to differ in.

Conjugate copies, and the number a table prints #

p6m has six maximal translationengleiche subgroups by count and four by kind. Three of them sit at index two and have one copy each — p6, p31m and p3m1. The fourth kind is cmm at index three, and there are three copies of it, one for each pair of mirror directions in the hexagonal cell. Conjugation by the sixfold rotation permutes them.

The International Tables print one entry with a multiplicity of three, because the three are the same subgroup differently placed, and that is exactly the distinction conjugation and sameness draws between an operation and its conjugates. Counting subgroups without deciding which question is being asked gives two different right answers: six maximal subgroups of p6m as subsets, four as conjugacy classes.

This site states both, because the two are used for different things. A domain count needs the number of copies — a crystal choosing between three orientations of cmm has three domain states. A classification needs the number of classes — there is one way for a hexagonal crystal to become orthorhombic, not three.

What the machinery did not decide #

Three limits, all of them real.

The klassengleiche half is enumerated only at index two and three. Those are the primes the quotient construction handles, and it is a genuine restriction rather than an oversight: a klassengleiche subgroup of index four can be maximal in ways index-two and index-three enumeration does not reach. What is claimed here is a theorem tested on a hundred subgroups of prime index, not a complete subgroup lattice.

The chains are translationengleiche only. A k-edge keeps the point group and enlarges the cell, so it very often lands on the same plane group again — p4 has a klassengleiche subgroup that is p4 on a doubled cell — and a chain following k-edges never has to stop. The eight chains counted above are the ones whose steps each throw an operation away, which is the descent a symmetry-lowering transition makes.

Nothing here says a transition happens. A group–subgroup relation says a descent is permitted, and this site’s applied field is careful about the difference: the descent of symmetry draws which classes can fall to which, and no essay in it predicts that anything will. Whether a crystal takes an available route is thermodynamics, and there is none of that here.

Who proved it, and what a chemist does with it #

Carl Hermann proved the theorem in 1929, in the same decade in which he was building the notation that carries half his name — and the two pieces of work are closer than they look, since both are about factoring a group’s information into parts that can be read independently. The theorem is stated for space groups in any dimension and the proof is short: if a subgroup H of G loses both an operation and a translation, then the group generated by H together with the parent’s full translation lattice sits strictly between them, so H was not maximal.

That single sentence is the whole of it, and it is worth noticing how little it uses. It needs the translations to form a normal subgroup of the space group — which they do, because conjugating a translation by any operation gives a translation — and nothing else. So the theorem is really a fact about groups with a normal abelian subgroup, wearing crystallographic clothes.

Hartmut Bärnighausen turned it into a working method in 1980. A Bärnighausen tree puts a parent structure at the top, its ordered or distorted derivatives below, and labels every edge with its kind and index: t2, k3, and so on. Reading such a tree tells a chemist which structures are related as parent and derivative, how many domain states each step permits, and which atomic positions split into how many independent ones — all before a single atomic coordinate is refined. The trees are in constant use in solid-state chemistry, and they are drawable only because every edge is of one kind.

The arithmetic in this essay is what a tree’s edges mean. What is added here is that both of the impossible edge labels — a step that keeps everything, and a step that keeps neither — are checked to be absent over a hundred instances rather than trusted to a theorem nobody in the room has read.

In three dimensions, where it is used #

The theorem is stated here for plane groups and proved for none: what has been done is a hundred instances, exhaustively, in two dimensions. Hermann proved it in 1929 for space groups in any dimension, and it is the reason the International Tables can list maximal subgroups at all — a table of all subgroups of a space group would be infinite, since the klassengleiche ones continue for every index, while the maximal ones are a short list closed under nothing.

In three dimensions the arithmetic is the same and the counts are larger. From seventeen to two hundred and thirty is where the step up happens, and the maximal-subgroup tables of the 230 are what a chemist consults when deciding whether a structure could be an ordered derivative of another. The rule that makes the consultation short is Hermann’s: two kinds of edge, and every descent factors.

The plane is where the theorem can be checked rather than cited, which is the reason for asking it here. Seventeen groups, a hundred maximal subgroups, two questions each, and both impossible answers empty. A hundred instances is not a proof and it is a good deal more than a restatement: an error in either question — a translation counted as present when it is absent, a linear part compared in the wrong basis — would fill one of the two empty cells immediately, and neither fills.

Where the ladder goes next #

The subgroups anchor now has two rungs: the index-two census, and maximality with the theorem that organises it. The obvious third is the other direction — supergroups, which are much harder, because a group has finitely many maximal subgroups and infinitely many minimal supergroups, and deciding which of them a given structure could sit inside is the question a structural chemist actually asks when looking for a missed symmetry.

The other direction is the normaliser, which decides when two descriptions of one group are the same description, and which turns out to be the object that counts the conjugate copies above. That is the next essay in this field.

The proof, which builds one group in the middle #

The theorem is used here and proved nowhere, and the argument is short enough to give — it constructs a single intermediate group and lets maximality do the rest.

Let H be a subgroup of G. Both have point groups: P_G and P_H, the sets of linear parts, with P_H a subgroup of P_G. Now define

M = the operations of G whose linear parts lie in P_H.

M is a subgroup of G, since the linear parts multiply, and H sits inside it — every operation of H has its linear part in P_H by definition.

Two facts about M are immediate. It contains every translation of G, because a translation has the identity as its linear part and the identity is in every point group — so M is translationengleiche in G. And it has the same point group as H, namely P_H, by construction — so H is klassengleiche in M.

So every subgroup sits in a chain H ⊆ M ⊆ G with a klassengleiche step and then a translationengleiche one. If H is maximal, there is nothing strictly between it and G, so M is one of the two ends: either M = G, and H is klassengleiche in G; or M = H, and H is translationengleiche in G.

That is the theorem, and it is worth noticing what it did not use. No property of the plane, no dimension, no lattice, no finiteness beyond the point group being a group. The same three lines prove it for space groups, for the subperiodic groups, and in any dimension.

The construction is also more useful than the theorem. M is the canonical intermediate group of any subgroup relation, maximal or not — so a descent losing both operations and translations decomposes into two named steps rather than into some pair a search has to find, and the group in the middle is the one a structural family will very often have a member sitting at.