One symmorphic group per class

The ten ways of being mm2 enumerates every consistent way of attaching translations to one point group on one lattice: sixteen assignments, ten groups after the quotient by origin shifts and axis relabellings. What that essay does not ask is whether one of the ten is special.

One is. Exactly one of them has an origin at which every operation’s translation part vanishes — where the group is the point group bolted straight onto the lattice, with no screw and no glide anywhere. It is Pmm2, and the corresponding statement holds for every class this site can enumerate.

What symmorphic means, arithmetically #

A space group’s operations are pairs (M, t): a linear part that the point group supplies, and a translation part that says where the operation acts and how far it slides. A space group is an extension is where that translation is split into two pieces — the intrinsic part, which belongs to the operation and no origin removes, and the locative part, which only records where somebody put the origin.

A group is symmorphic when there is a single point at which every linear part can be realised with no translation: choose it as the origin, and for each M in the point group the group contains an operation (M, t) whose t is a lattice vector. On a primitive lattice that is the same as saying every operation’s intrinsic part is zero, and the definition can then be said in one more way — a primitive symmorphic group has no screws and no glides. The qualification is not decoration, and a later section is entirely about what goes wrong without it.

The test is an origin shift tried everywhere. Moving the origin by s sends t to t + (I − M)s, so the question is whether some s clears every translation part at once — and it is asked on a grid fine enough to carry the denominators the operations have, since a threefold’s intrinsic translation lives in thirds and a fourfold’s in quarters.

One in each class, and why that is the interesting statement #

Running that test over every group each arithmetic class produces gives one symmorphic group in each. Not “usually one” and not “at least one”: exactly one, in every class the site enumerates, over twenty-five groups in six classes.

Both halves are worth separating.

At least one, always. Given a point group P and a lattice Λ, the set of pairs (M, 0) for M in P, together with all lattice translations, is a group — the composition of two such operations is (M₁M₂, M₁t₂ + t₁) and every translation involved is a lattice vector. So the semidirect product of the point group with the lattice always exists, and it is symmorphic by construction. There is nothing to check: a symmorphic group is available in every class because it can be written down.

At most one. Two symmorphic groups in the same class would be two ways of bolting the same point group onto the same lattice, differing by something other than an origin shift — and there is nothing for them to differ by, since every operation is (M, 0) at the common origin and the linear parts are the same set. So the two coincide.

The bijection follows: arithmetic crystal classes and symmorphic space groups are the same objects counted twice. There are seventy-three of the classes in three dimensions and seventy-three symmorphic groups among the two hundred and thirty, and the equality is not a coincidence to be remembered but a definition unpacked.

What the other groups are #

The non-symmorphic groups in a primitive class are the ones with a genuine screw or glide, and there are between one and nine of them depending on how much room the point group leaves. The primitive orthorhombic class mm2 has ten groups: Pmm2 symmorphic, and nine with a glide or a screw in some direction.

That count of ten is the one the ten ways of being mm2 arrives at, and its relation to the symmorphic one is the whole content of the extension problem. Attaching translations to a point group is choosing a cocycle; changing the origin is changing it by a coboundary; the distinct groups are the cohomology classes; and the symmorphic group is the class of zero. Every other group in the class is a non-trivial element of a finite group of extensions, and the size of that group is what makes some classes rich and others poor.

Two numbers make the point across the whole of the two hundred and thirty. Seventy-three groups are symmorphic. The remaining hundred and fifty-seven are not, so more than two thirds of all space groups contain an operation that no origin can straighten — which is a striking figure for a construction whose intuitive picture is “a point group repeated on a lattice”.

The closure is what makes it non-trivial #

There is a reason the test has to be a search rather than an inspection, and what the closure adds is the whole of it.

A group is not a list of generators. Choose translations for two generators, close the set under composition, and operations arrive that neither generator contained — a screw where there were only mirrors, a glide where there was a mirror and a centring vector. So a group written down with two clean generators can be non-symmorphic anyway, because the closure produced an intrinsic translation that neither of the chosen operations had.

That is why “does this group have a screw or a glide” cannot be answered by looking at the symbol’s letters alone, and why the test here is an origin search over the whole operation set. The symbol reports the generators the convention names; the group is everything the closure produced.

The converse trap is the same one from the other side. A group written with a screw generator may be symmorphic, if a shift of origin turns that screw into a plain rotation — which happens exactly when the screw’s translation is entirely locative. Deciding needs the split into intrinsic and locative parts, and that split is a computation rather than a reading.

The sentence that is true on a primitive lattice and false on a centred one #

“A symmorphic group has no screws and no glides” is what everybody learns, and it is wrong for a third of the lattices.

Take C2. It has a two-fold axis along b at x = 0, and it has the C-centring vector (½, ½, 0). Compose them: the product is a two-fold whose translation part is (½, ½, 0), whose component along the axis is (0, ½, 0), and which is therefore a 2₁ screw — sitting at x = ¼, exactly where the International Tables draw it. That screw is a genuine operation of C2. And C2 is symmorphic: it is one of the seventy-three, and it has to be, since it is the point group 2 bolted straight onto a C-centred lattice and nothing else.

The two statements are compatible because symmorphy is a question about cosets rather than about operations. For each linear part M, the group contains a whole coset of operations (M, t + λ) as λ runs over the lattice; symmorphy asks whether some member of each coset loses its translation at one common origin, not whether every member does. On a primitive lattice each coset gives one operation per cell and the two questions coincide. On a centred lattice a coset gives several, and the plain rotation and the screw it generates are two of them.

The extreme case in the table is Fm3̅m — face-centred, symmorphic, and containing forty-five screw axes and twenty-four glide planes. Nobody would describe that group as free of screws, and nobody has to: what makes it symmorphic is that its forty-eight linear parts are all available at the cell origin with nothing but lattice vectors attached.

Why it matters which kind a structure is in #

Three consequences, and the first is the one that shows up in an experiment.

Systematic absences. Systematic absences are what a screw or a glide does to the data, and they are the reason the distinction is experimental rather than merely descriptive. A screw or a glide extinguishes a whole class of reflections, exactly, for every possible arrangement of atoms — which is reading a space group from its absences. A symmorphic group extinguishes nothing of its own, and it gets there two different ways: on a primitive lattice because it has no screw and no glide at all, and on a centred one because the screws and glides the centring generates cancel exactly the reflections the centring rule has already cancelled. C2’s screw axis forbids 0k0 with k odd, and the C rule forbids every reflection with h + k odd, which on the 0k0 row is the same set. So a diffraction pattern with a primitive lattice and no absences at all is the signature of a symmorphic group, and on a centred lattice the signature is a pattern with the centring rule and nothing else.

That is also why symmorphic groups are the hardest to identify. Absences narrow the possibilities sharply and a group with none gives the least information, so a primitive symmorphic group has to be told from its neighbours by intensity statistics rather than by extinctions.

Site symmetries. In a symmorphic group there is a point at which the full point group acts, so an atom placed there has the whole class as its site symmetry. In a non-symmorphic group there is no such point: the operations do not all pass through anything, so the largest site symmetry is a proper subgroup of the class. A molecule with the full symmetry of the crystal class can therefore sit in a symmorphic group and cannot sit in a non-symmorphic one — which is a genuine chemical constraint and not a matter of description.

Representation theory. The irreducible representations of a symmorphic group at any wavevector are those of the point group of that wavevector, straightforwardly. In a non-symmorphic group they are projective representations, and at the zone boundary the bands are forced to stick together in pairs — a degeneracy that comes from the group’s structure rather than from any interaction. Physicists meet non-symmorphy first as an unavoidable band degeneracy.

The classes, and why they are arithmetic #

The word doing the work in the bijection is arithmetic, and it means something specific.

A geometric crystal class is a point group up to conjugacy: the thirty-two. An arithmetic crystal class is a point group together with the lattice it acts on, up to a change of basis: the same point group on a primitive and on a centred lattice are different arithmetic classes, because no change of basis turns one into the other. There are seventy-three of them, and the increase from thirty-two comes exactly from the centring choices the fourteen lattices offer each point group.

That is why the bijection is with arithmetic classes rather than with geometric ones. Bolting mm2 onto a primitive orthorhombic lattice gives Pmm2; bolting it onto a C-centred one gives Cmm2; onto an F-centred one, Fmm2. Three symmorphic groups, three arithmetic classes, one geometric class — and the count of seventy-three on both sides tracks the lattices rather than the point groups.

Reading the group as a product #

There is a structural way of saying what a symmorphic group is, and it is the one that makes the bijection obvious rather than merely true.

Every space group has its translations as a normal subgroup, and the quotient by them is the point group — that much holds for all two hundred and thirty. What distinguishes the symmorphic ones is whether that quotient splits: whether the point group can be found sitting inside the space group as a subgroup, rather than only as a quotient of it.

In a symmorphic group it can. Take the origin that clears every translation, and the operations (M, 0) form a copy of the point group inside the space group, meeting the translations only at the identity. The space group is then the semidirect product of the lattice by the point group, written Λ ⋊ P, and everything about it is determined by the pair.

In a non-symmorphic group it cannot. Every copy of a screw’s linear part inside the group carries a translation that no conjugation removes, so there is no subgroup isomorphic to the point group at all — the extension is non-split, and the group carries information beyond the pair (Λ, P).

So symmorphic means split, and the bijection reads: one arithmetic class is one pair (Λ, P), one pair gives one split extension, and one split extension is one symmorphic group. Three statements of the same fact, and the last is the one the count of seventy-three refers to.

The zero element, which is what “one in each class” means #

The bijection is proved above by two arguments — at least one, at most one — and there is a single statement containing both, in the vocabulary the extension count uses.

The ways of attaching translations to a point group, up to an origin shift, form a group: the second cohomology of the point group with coefficients in the lattice. Adding two attachments gives an attachment; the origin shifts are what is quotiented out; and the result is a finite abelian group whose elements are the space groups of that arithmetic class, before the normaliser’s merges.

The symmorphic group is the zero element. Attaching nothing at all is a legitimate attachment, it satisfies the cocycle condition trivially, and its class is the identity of that group.

So at least one is the statement that a group has an identity, and at most one is the statement that it has only one — both of which are true of every group whatever, which is why the bijection needs no case analysis and holds in every dimension.

And the count of non-symmorphic groups in a class is the number of non-zero classes, after the normaliser identifications. That is why a class with a large cohomology has many groups and one with trivial cohomology has exactly the symmorphic one — nine of the thirteen plane classes being of the second kind, which is where thirteen of the seventeen come from.

Seventy-three of two hundred and thirty, and what a database shows #

The count has a distribution worth reading, and the reading contradicts what the arithmetic alone would suggest.

Seventy-three of the two hundred and thirty are symmorphic — a little under a third — so if space groups occurred uniformly, a third of published structures would be in one.

They do not. The commonest space groups in the small-molecule literature are P2₁/c, P1̅, C2/c, P2₁2₁2₁ and P2₁, and of those only P1̅ is symmorphic. The overwhelming majority of published organic structures are non-symmorphic, and the reason is packing rather than symmetry: screws and glides are the operations that pack a molecule against itself efficiently, and mirrors — which the symmorphic groups are full of — are the ones that pack badly.

So the two counts pull in opposite directions. Symmorphic groups are a third of the classification and a small minority of the structures, and the gap is a measurement about molecules rather than a fact about groups.

It has one practical consequence. Since a screw or a glide produces systematic absences and a symmorphic group produces none but the centring’s, the commonest groups are the ones a diffraction pattern identifies most easily — and the ones the arithmetic calls simplest are the ones an experiment finds hardest to pin down.

What this computation covers #

The honest scope, stated plainly.

Six arithmetic classes are enumerated here in full, and they produce twenty-five groups between them. Those are the classes whose enumeration is small enough to run in full every time the figure is drawn: two monoclinic, one orthorhombic pair, the primitive tetragonal fourfold and the primitive trigonal threefold. In each, the number of symmorphic groups is computed and asserted to be one.

Seventy-three is quoted. The count of arithmetic classes in three dimensions, and hence of symmorphic groups, comes from the literature and is named as such. What is demonstrated here is the mechanism — that the count on each side is one per class — rather than the total.

The origin search is on a grid of twelfths. Every intrinsic translation in the site’s groups has a denominator dividing twelve, so the search is exact for them. A finer grid finds nothing new, which is checked; a coarser one would report a symmorphic group as non-symmorphic, which is the failure worth naming because it errs on the side of a false negative and would be easy to miss.

How common each kind actually is #

The counting above says seventy-three of two hundred and thirty are symmorphic. What structures do is a different question, and the answer is worth putting beside the arithmetic because it points the other way.

The space groups that real crystals occupy are heavily concentrated in a handful of entries, and the most populated of them are non-symmorphic: P2₁/c alone accounts for a third of all published organic structures, and P2₁2₁2₁ for most of the rest of the chiral ones. Both have screws in every direction and neither is symmorphic.

The reason usually given is packing. A screw axis or a glide plane moves a molecule along its own axis as well as turning or reflecting it, so it stacks molecules in an offset arrangement — which is how an irregular shape packs densely. A pure rotation stacks copies face to face, which is efficient for a symmetric molecule and wasteful for a lumpy one. So symmorphic groups suit high-symmetry inorganic structures and non-symmorphic ones suit molecules, and the counts in the structural databases show it clearly.

That is an empirical statement about materials and nothing in this essay derives it. What the arithmetic does say is that the option exists in every class: whatever the point group and lattice, there is exactly one way to build the group with no screws and no glides, and every crystal that takes another way has taken one of finitely many alternatives that this enumeration counts.

Where the ladder goes next #

The space-groups anchor now has six rungs: the step from seventeen to two hundred and thirty, the round trip in space, the extension picture, one class enumerated in full, the enantiomorphic pairs, and the symmorphic ones.

The rung above is the sixty-five for a chiral molecule — the Sohncke groups, those with no operation of negative determinant, which are the only groups a protein or any other single-handed molecule can crystallise in. That count needs an enumeration over all two hundred and thirty rather than over six classes, so taking it would mean either building the full table or being explicit that the number is quoted.

The rung sideways is the cohomology this essay gestures at. The extensions of a point group by a lattice form a finite abelian group, its order is computable, and the number of space groups in an arithmetic class is the number of its orbits under the normaliser. Making that concrete — computing the extension group for one class and reading the ten off it — would replace an enumeration with a structure, and it is the natural thing to do next with machinery this site already has.