Six ways to name one group

One group, three symbols shows a monoclinic group wearing three names — P2₁/c, P2₁/a and P2₁/n — because the six standard cell choices with b unique give three distinct spellings of one thing. This essay does the same arithmetic in the orthorhombic system, where the ambiguity is at its worst and the reason for it is different.

In a monoclinic cell one axis is picked out by the two-fold and the other two are free to be redescribed. In an orthorhombic cell all three axes are inequivalent and nothing about the lattice says which is which. Naming them is a choice, and there are six ways of making it.

Pnma, and its five other names #

Pnma is number 62 and is one of the most common space groups in inorganic crystallography — perovskites, olivines and a great many oxides are in it. Its six settings are

Pnma · Pmnb · Pbnm · Pcmn · Pmcn · Pnam

and a reader meeting Pbnm in a paper has no reason to suspect it is Pnma unless they already know. The mineralogical literature on perovskites used Pbnm for decades; the modern convention is Pnma; and the coordinates in the two are related by a permutation, so a structure copied from one paper into another without transforming them is wrong in a way that looks plausible.

The same holds across the system. Pmc2₁ is also Pcm2₁, P2₁ma, P2₁am, Pb2₁m and Pm2₁b. Pna2₁ has six; Pmm2 has three, because two of its directions are equivalent and the permutations that swap them produce the same symbol; Pbca has two, because its own symbol is invariant under a three-cycle of the axes.

That last case is worth pausing on: the number of settings a group has is the number of distinct symbols, not six, and how many distinct symbols there are is decided by which permutations the group’s own operations are invariant under. It is an orbit count, and it varies from one to six across the system.

Nothing here is a table of alternative symbols #

Each setting is produced by conjugating the group’s own operations by the change of basis, and its symbol is then derived from those operations by the same routine that has to reproduce every group’s entered symbol for the site’s gate to pass.

So a spelling appears here because the operations put it there. Pnma’s b glide becomes an a glide in a permuted basis not because a table says so but because the operation whose translation was b/2 now has a translation along the axis that is being called a.

The check that these are settings rather than different groups is made while each is computed, and it has three parts. The change of basis has determinant ±1, so the lattice is preserved. Conjugating back reproduces the original operation set exactly. And the census of screws, glides, mirrors and rotations is unchanged, since conjugation cannot turn one kind of operation into another. A symbol that differs while all three hold is a second name for one group, which is the whole subject.

The permutations have to be made proper, and that is load-bearing #

A permutation of three axes has determinant ±1, and the three odd permutations have determinant −1. An improper change of basis turns a right-handed cell into a left-handed one, and that is not a relabelling — it is a reflection.

The consequence is specific and would be easy to miss: an improper permutation turns a 4₁ screw into a 4₃, so a group would be reported as a setting of its own enantiomorphic partner. The two are different groups, two hundred and thirty or two hundred and nineteen is about exactly that distinction, and a settings census that silently identified them would be destroying the distinction it exists to preserve.

So each odd permutation carries a sign on one axis — the Tables write them bac̅, c̅ba and ac̅b, and the overbar is precisely that correction. With the signs in place every change of basis has determinant +1 and the enantiomorph question does not arise.

The refusal is the other half. A change of basis that is not a permutation — doubling an axis, say — has determinant two, and the machinery refuses it at a lower level than the settings check: the inverse of a matrix with determinant two is not integral, so the conjugation cannot be performed at all. Doubling an axis describes a sublattice, and the group of the sublattice is a subgroup of the original rather than another name for it.

Where the ambiguity is, and where it is not #

A tetragonal, trigonal, hexagonal or cubic group has one name under these permutations, and the reason is worth stating because it is not a convention.

Those systems have a principal axis picked out by the group’s own operations — the four-fold, the three-fold, the six-fold, or the set of four three-folds in the cubic case. Once the operations distinguish an axis, calling it c is not a choice; it is a reading. There is nothing left to permute.

Orthorhombic and monoclinic groups have no such axis. An orthorhombic group’s three two-folds are mutually perpendicular and indistinguishable by any property of the group, so the labelling is free and the symbol follows the labelling.

That is the shape of the International Tables’ 530 settings of 230 groups in miniature. The excess of 300 is not spread over the groups; it sits almost entirely in two systems, and inside those systems it sits on the groups whose symbols are not invariant under axis permutations. The forty-five groups defined here give ninety-five settings between them, with every one of the fifty extra names in the same two systems.

What a setting is not #

Three things get called settings and only one of them is.

A change of origin is not a setting, and it is two origins for one group. Fddd has two published origins an eighth apart, both describing the same group with the same symbol, and choosing between them changes every coordinate without changing a letter.

A change of cell is not a setting either. A rhombohedral group can be described on hexagonal axes or on rhombohedral ones, and the two cells have different volumes — a factor of three — so this is a change of lattice description rather than a relabelling. The Tables list both and call them settings; the determinant is not ±1, and by the criterion used here they are something else.

And a symbol difference from the e glide is not a setting. The plane that carries two glides is a case where one plane holds two operations and the pre-1992 symbol chose one, so Cmca and Cmcb were two names for one group in one setting. That ambiguity was removed by inventing a letter; this one cannot be, because there is nothing arbitrary to remove — the axes really are unlabelled.

What that looks like on a diagram is worth holding in mind, because it is the reason the symbol moves and the group does not. Draw Pnma’s plan and every element sits somewhere: the n glide perpendicular to one axis, the mirror perpendicular to another, the a glide perpendicular to the third, with screw axes running between them. Permute the axes and the same elements are still there, in the same relative arrangement, viewed down a different direction — so the diagram is redrawn rather than changed, and the symbol, which reads positions off directions in a fixed order, comes out spelled differently. A reader who has the plan in front of them can see that nothing about the group moved. A reader who has only the symbol cannot.

The monoclinic case, for comparison #

Setting the monoclinic case beside the orthorhombic one shows that the two ambiguities have different sources, which is worth knowing because the fixes are different.

In a monoclinic group the unique axis is determined — it is the direction of the two-fold, or the normal of the mirror — so there is no freedom in which axis is called b. The freedom is in the other two, which span a plane and can be redescribed by any basis of that plane. The Tables restrict to six standard choices and P2₁/c picks up three distinct names from them: P2₁/c, P2₁/a, P2₁/n.

The glide letter is what changes, because the glide’s translation is a fixed vector and which axis it is called depends on the basis. The n in P2₁/n is a diagonal glide only because the two in-plane axes were chosen so that the translation lies along their sum.

So the orthorhombic ambiguity is about which axis is which and the monoclinic one is about which basis spans a plane. Both are relabellings; only the first is a permutation.

Where the exactness stops #

The conjugation is exact, in integers and rationals, and so is the comparison that says the operation set came back.

The symbol derivation is a convention faithfully reproduced. Every symbol here is derived from operations rather than looked up, and the agreement with the Tables says the derivation reproduces the Tables’ convention — not that the convention is the only reasonable one.

The count of settings is over the groups defined here, which is forty-five of two hundred and thirty. The number 530 is quoted from the Tables and named as quoted; what is computed is which of these groups have how many, and the answer is ninety-five for forty-five.

And the standard setting is a choice with no mathematics behind it. The Tables’ choice of Pnma over Pbnm is a convention adopted for consistency, and a paper using the other one is not making an error. What is an error is quoting coordinates from one setting under the symbol of another, and it happens often enough that structure databases carry transformation code for it.

Reading a symbol against the wrong setting #

The practical hazard is worth spelling out, because it is the one this whole rung exists to prevent.

A Hermann–Mauguin symbol is positional: the letters correspond to directions in a fixed order, which for orthorhombic groups is a, b, c. So Pnma says there is an n glide perpendicular to a, a mirror perpendicular to b, and an a glide perpendicular to c — and Pbnm says a b glide perpendicular to a, an n glide perpendicular to b, and a mirror perpendicular to c.

Those describe the same group in different bases, and reading either symbol without knowing which basis is meant gives a definite and wrong answer about where the mirror is. A symbol is not a name; it is a description relative to axes, and reading Hermann–Mauguin is the whole of that argument at the plane-group level.

The failure mode is not that a reader gets confused. It is that they get a consistent and incorrect structure: coordinates from one setting refined against a symbol from another produce a model that refines, that has plausible bond lengths, and that has the mirror in the wrong place.

Who standardised it, and when #

The proliferation is older than the standard. Hermann and Mauguin’s notation dates from 1928–31, and the International Tables for X-ray Crystallography of 1952 was the first to give a single recommended symbol per group along with the alternatives.

The 530 settings are the 1983 edition’s count, and the number is a deliberate compromise: it lists all six orthorhombic permutations for every group, the monoclinic cell choices, and both origins where a group has two, but not every conceivable basis change. That is why 530 rather than an unbounded number — the Tables list the settings a crystallographer will meet, not the ones that exist.

The mineralogical use of Pbnm for perovskites persisted well past the standardisation, and the two literatures coexisted for thirty years. The transformation between them is a permutation and a cyclic shift of coordinates, which is one line, and the number of published structures that got it wrong is not small.

What a settings census is good for #

A census of settings is not bookkeeping. It answers three questions that come up whenever symbols are compared by machine.

Are these two symbols the same group? Compute both groups’ settings and intersect. That is more reliable than a lookup table, because a lookup table has to be maintained and a derivation does not.

Which symbol is standard? The Tables’ choice, which this site records rather than derives — and the census is what says how many alternatives a reader might meet.

How much does a symbol constrain? A group with six distinct symbols has a symbol that carries the full information about which direction holds what; a group with one has a symbol invariant under relabelling, which means its own operations already fix the labelling. So the count of settings is a measure of how much of a group’s structure the axes carry rather than the operations — and Pbca having two rather than six says something about Pbca that no single symbol does.

What the six permutations do to the coordinates #

A setting change is a relabelling of axes, and a relabelling of axes moves every atom’s coordinate triple, which is the part that does damage in practice.

The transformation is the same matrix that conjugated the operations, applied to positions. Going from Pnma to Pbnm is the cyclic permutation (a, b, c) → (b, c, a), so an atom at (x, y, z) in the first is at (y, z, x) in the second — and every published table of coordinates is in one convention or the other with nothing on the page to say which.

The check that catches a mistake is a bond length. Coordinates transformed wrongly give a structure whose interatomic distances are physically absurd — a metal-oxygen bond of 1.1 Å, or two atoms on top of each other — and any refinement will report it immediately. So the error is loud when it is made in full and quiet when it is made in part: transforming the coordinates and forgetting the anisotropic displacement parameters, which are also tensors and also permute, gives a structure that refines with slightly wrong thermal ellipsoids and no other symptom.

That is the practical reason a settings census is worth having as machinery rather than as a table: the transformation is mechanical, and anything mechanical done by hand is done wrongly eventually.

The indices are relabelled too, and that is where it bites first #

The transformation of coordinates is the part that produces absurd bond lengths and is therefore caught. There is a second transformation that produces nothing absurd at all, and it is the one a reader is more likely to meet.

A permutation of the axes permutes the reciprocal axes in the same way, so every reflection index is relabelled with the axes. A reflection that is 0 2 0 in one setting is 0 0 2 in another, and a list of measured intensities carried across a setting change without its indices being transformed is a list describing a different crystal.

The consequence shows up in the one place a setting is most often quoted: the absence conditions. A systematic absence is a statement about indices — 0kl absent for k + l odd, h00 absent for h odd — so the same operation produces a differently-worded condition in each of the six settings. The n glide of Pnma is perpendicular to a and its condition is on 0kl; the same operation in Pbnm is perpendicular to c and its condition is on hk0. Nothing about the crystal has changed and both conditions are correct.

That is why reading a group off its absences is done against a stated setting rather than against a list of conditions memorised in one. A determination that matches its observed absences to Pnma’s table while the data were indexed on Pbnm’s cell will find no group consistent with the data, and the failure gives no indication of its cause — the conditions simply do not fit, which is exactly what a wrong group looks like.

The repair is the same as everywhere in this rung: transform, then compare. The permutation that conjugates the operations is the one that permutes the indices, it is the same integer matrix, and applying it to both at once is the only arrangement in which nothing has to be remembered. The reciprocal basis transforms by the inverse transpose, which for a signed permutation is the permutation itself — the one case where the distinction can be ignored, and the reason it is so easy to forget that there is one.

Where the ladder goes next #

This rung establishes what a setting is and where the ambiguity lives. Two rungs sit above.

The full 530. Enumerating every setting of every group — the six permutations, the monoclinic cell choices, the two origins, the rhombohedral axes — and checking each against the Tables’ list is a computation this site could make once it defines all 230 groups. It would be the strongest possible check on the symbol derivation, because it asks the derivation to reproduce not forty-five names but five hundred and thirty.

Transformations between settings, as a service. Given a structure in one setting and a target setting, produce the transformation of coordinates. That is the operation databases perform and papers get wrong, and it is one matrix — the same matrix that conjugated the operations here, applied to positions instead.