Thirty-two classes, eighteen groups

Thirty-two crystal classes and no others is the count this subject is organised around. It is a count of actions: a class is a group of matrices acting on space, and two classes are different when no change of basis carries one onto the other.

Ask a weaker question — forget the matrices and keep only which product is which — and the count changes.

Eighteen.

What collapses #

The first row that carries more than one class is the smallest interesting object in crystallography.

1̅, 2 and m are the same group. An inversion centre, a two-fold rotation and a mirror plane: three operations that could hardly be more different in what they do to space, each generating a group with two elements — the identity and itself. Their multiplication tables are identical because there is only one group of order two.

2/m, 222 and mm2 are the same group. Four elements, every one of order two, every product of two distinct non-identity elements the third: the Klein four-group. One of them has a two-fold axis with a mirror across it, one has three perpendicular two-fold axes, one has an axis with two mirrors through it. The tables do not distinguish them.

32 and 3m are both the dihedral group of order six, which is also the symmetric group on three letters. 3̅, 6 and 6̅ are all cyclic of order six. And the largest collision is 3̅m, 622, 6mm and 6̅2m, four hexagonal classes that are one dihedral group of order twelve.

The eighteen, listed #

Reading the table by order gives the whole answer in a paragraph.

Order 1: the trivial group, realised by the class 1. Order 2: C₂, realised three ways — 1̅, 2, m. Order 3: C₃, realised once, by 3. Order 4: C₄ (by 4 and 4̅) and C₂ × C₂ (by 2/m, 222, mm2). Order 6: C₆ (by 3̅, 6, 6̅) and D₃ (by 32 and 3m). Order 8: C₄ × C₂ (4/m), C₂ × C₂ × C₂ (mmm), and D₄ (422, 4mm, 4̅2m). Order 12: C₆ × C₂ (6/m), D₆ (3̅m, 622, 6mm, 6̅2m), and A₄ (23) — the alternating group on four letters, which is the rotation group of a tetrahedron. Order 16: D₄ × C₂ (4/mmm). Order 24: A₄ × C₂ (m3̅), D₆ × C₂ (6/mmm), and S₄ (432, 4̅3m) — the symmetric group on four letters, the rotation group of a cube. Order 48: S₄ × C₂ (m3̅m).

Two patterns are worth noticing. The centrosymmetric classes are all direct products with C₂, because adding an inversion centre to a group with no inversion in it multiplies it by the group of order two — the inversion commutes with everything. And the classes that appear alone in their row are mostly the ones with an inversion, for the same reason: attaching a centre to a group pins it down.

How the merge is computed #

Nothing here is looked up. Two groups are isomorphic when some bijection preserves multiplication, and the search for one is finite.

Take a generating set of the first group — this collection already computes one, and it is one, two or three elements for every crystal class. Each generator may go to any element of the second group of the same order, since an isomorphism preserves order. Extend the assignment: every element of the first group is a word in the generators, so its image is the same word in the images, and the extension either is well defined or is not — a word reached two ways must land in the same place. Check the result is a bijection onto the whole of the second group.

The search runs over the product of the choices, which for these groups is at most a few thousand tuples, and cheap invariants throw out most pairs before it starts: the order, the multiset of element orders, whether the group is abelian, the size of the centre, and the number of conjugacy classes.

What is deliberately not required is that traces agree. This collection already has the stronger test — a character-preserving isomorphism, which is the condition for two matrix groups to be conjugate. Dropping the trace condition is the entire content of the question, because a mirror has trace 1 and a two-fold rotation has trace −1, and the two generate the same abstract group.

Why the classification keeps them apart anyway #

An obvious reaction to eighteen is that the thirty-two are over-counting. They are not, and the reason is the whole justification for the classification.

A crystal’s physical properties respond to the action, not to the table. Neumann’s principle says a property tensor must be invariant under the point group, and invariance is a statement about matrices: which components of a tensor survive depends on how the group acts on space, not on how its elements multiply.

The three order-two classes make this concrete. 1̅ permits no piezoelectricity and no pyroelectricity. 2 permits both, with the polar direction along the axis. m permits both, with the polar direction in the plane. Same group, three completely different lists of permitted properties.

The same holds for the four-element classes. mm2 is polar and 222 and 2/m are not; 2/m is centrosymmetric and the other two are not; 222 is chiral and mm2 is not. Three classes, one group, and every question a crystallographer asks separates them.

So the eighteen is not a better count. It is a count of a different thing, and the interesting fact is how much information the passage from thirty-two to eighteen throws away: everything about orientation, everything about handedness, everything about which operations are proper.

The two ways a classification can be coarse #

It is worth separating two things that both sound like “the same group” and are not.

Conjugate as matrix groups. Two sets of matrices related by a change of basis: P G P⁻¹ = G′. This is the relation the thirty-two are counted up to, and this collection computes it by finding the conjugating matrix rather than by comparing lists.

Isomorphic as abstract groups. Two groups with a multiplication-preserving bijection between them, with no requirement that it come from a change of basis.

Conjugate implies isomorphic and not conversely, and the gap between them is exactly the thirty-two-to-eighteen collapse. What the gap contains is the information a representation carries beyond the group: 222 and mm2 are the same group with two different three-dimensional representations, and choosing between them is choosing how the group acts rather than which group it is.

That distinction runs through the whole of this subject in one direction or another. The ten geometric and thirteen arithmetic classes in the plane differ for the same reason, one level down: an arithmetic class is a group of integer matrices up to change of integer basis, a geometric class allows any basis, and the three splittings are three groups holding a lattice in two ways.

It is worth saying where the thirty-two itself comes from, because the two counts are produced by the same machinery run with one clause changed. The classes are found by closing sets of integer matrices inside the two holohedries and then merging whatever a change of basis identifies — the merge is a search for a matrix P with P·G·P⁻¹ = G′, and it either produces one or the pair stays apart. The eighteen is that same list merged again under a weaker test: not “is there a change of basis”, but “is there any bijection at all preserving multiplication”. Every merge the first test makes, the second makes too; the fourteen extra merges are the whole content of the collapse, and they are the pairs where a bijection exists and no basis carries it.

What the abstract type does decide #

It is not useless. Three things are decided by the multiplication table alone.

The character table, and therefore the number of irreducible representations and their dimensions. Isomorphic groups have the same character table as an abstract object — though the assignment of which representation describes a given physical quantity depends on the action, which is why a character does not know its basis.

The subgroup structure, as a lattice of abstract groups: which orders occur, how many subgroups of each, which are normal. That is what makes 2/m, 222 and mm2 all have exactly three subgroups of order two.

The number of conjugacy classes, which is the number of irreducible representations, which is the number of distinct symmetry species a vibration or an orbital can have. A spectroscopist’s count of bands is an abstract-group count; which band is which is not.

There is a third cut worth naming here, because a reader who has just met two counts will meet a third within a page and should know it is not a rival to either. The seven crystal systems sort the thirty-two by the lattice a class can sit on, which is coarser than the classes and organised by something neither the matrices nor the multiplication table decides on its own. 2/m, 222 and mm2 are one abstract group and two systems; 4 and 4̅ are one abstract group and one system. So the three classifications cut the same list three ways, and no two of them are nested — which is the general shape this essay is about, and the reason a count without its equivalence relation is not a fact.

The three order-two classes, in full #

The smallest collapse deserves the most attention, because everything the essay is about is visible in it and nothing else is.

1̅, 2 and m each consist of the identity and one other operation of order two. As abstract groups they are indistinguishable — there is exactly one group of order two, and every group of order two is it.

As actions they differ in the most basic way available: by how many directions they leave fixed.

The inversion 1̅ fixes one point and no direction. Its matrix is −I, its determinant is −1, its trace is −3. It reverses every vector, so no direction survives it.

The rotation 2 fixes a line. Its matrix has determinant +1 and trace −1, and its eigenvector of eigenvalue +1 is the axis. It is a proper motion — a crystal with only this symmetry is chiral and can rotate light.

The mirror m fixes a plane. Determinant −1, trace +1, and a two-dimensional fixed space. It reverses handedness, so a crystal with a mirror is not chiral.

Three fixed spaces of dimension zero, one and two, three different determinants, three different traces, three completely different lists of permitted physical properties, and one multiplication table. The table records that each operation is its own inverse and nothing else, which is true of all three and is all they have in common.

What the round trip checked, and how #

Groups of different order must never be identified, which is the trivial test and catches a search that returned a partial map.

Same order and different table must not merge. The class 4 is cyclic of order four; 222 has three elements of order two and no element of order four. Both have four elements and they are not the same group.

A mirror and a two-fold rotation must be identified. This is the test that says which classification is being computed: a stricter test that kept them apart would return thirty-two and would be measuring conjugacy of matrix groups, which is a question already answered elsewhere on this site.

And two order-eight groups that differ must not merge: 422 is dihedral and 4/m is abelian, and a search that found a “bijection” between them would have failed to check that products are preserved.

Where a spectroscopist meets this #

The place the eighteen matter in practice is molecular and solid-state spectroscopy, and it is worth saying how.

Vibrations and orbitals are labelled by irreducible representations. How many kinds there are is the number of conjugacy classes, which is an abstract-group quantity — so 222 and mm2 have the same number of symmetry species (four), the same dimensions (all one), and the same character table as an abstract object.

Which species is which is not. The species of mm2 are labelled A₁, A₂, B₁, B₂ and the A₁ one contains a polar vector along the axis, so mm2 permits a dipole and is pyroelectric. The species of 222 have the same names and none of them contains a polar vector, so 222 is not. The table is the same and the assignment of physics to rows is different.

That is the practical form of the distinction this essay is about: counting bands is an abstract-group question and assigning them is not. A spectroscopist reading a character table is using both halves and the tables print both, which is why the character table of a point group carries a column of “functions transforming as” that the abstract group cannot supply.

Where the exactness stops #

The three-dimensional case is the one computed; the plane is different. The ten plane point groups realise how many abstract groups is a smaller question with a smaller answer, and it is not asked here, since the plane’s classification is carried elsewhere in this collection and its point groups are drawn rather than tabulated. Nothing about the merge transfers between dimensions, because which classes exist changes.

Eighteen is a count of isomorphism types among the thirty-two, not of all groups of those orders. There are two abstract groups of order four and the classes realise both; there are five of order eight and the classes realise only three. The eighteen is what crystallography happens to reach, and nothing here says which abstract groups are missing or why.

The search is complete for these groups and is not a general algorithm. Deciding whether two arbitrary finite groups are isomorphic is a genuinely hard computational problem — no polynomial algorithm is known for the general case — and what makes it easy here is that the groups have at most forty-eight elements and small generating sets.

The names are read off, and the reading is not a proof. A group whose largest element order equals its order is cyclic; an abelian group of exponent two is a product of copies of C₂. Those inferences are correct and the machinery makes them; a name like “D₆ × C₂” is a description of the structure computed rather than an identification against a database of groups.

The invariants used to narrow the search are not a complete invariant. Two non-isomorphic groups can share order, element orders, abelianness, centre size and class count — the smallest examples have order sixteen — so the invariants only skip work and the isomorphism is always decided by an actual bijection. That is worth stating because a merge based on invariants alone would be plausible, fast and wrong.

And the merge says nothing about space groups. Two space groups with isomorphic point groups are not thereby related, and the two hundred and thirty do not collapse in any comparable way — most of them are not isomorphic as abstract groups at all, since their translation subgroups differ.

Who counted them, and when #

The abstract classification of the crystallographic point groups is not a landmark result and has no single author, which is itself informative: the merge is easy once the classes are known, and knowing the classes is the hard part.

What the nineteenth century did care about was the reverse question — which abstract groups can act crystallographically at all — and that is the restriction plus a good deal of case analysis. Hessel produced the thirty-two in 1830 and was ignored; Gadolin rediscovered them in 1867. Neither had the language of abstract groups: Hessel’s argument is about which combinations of axes and planes can coexist, and the word “group” in its modern sense was not yet in use for anything but permutations.

The habit of separating a group from its representation is Frobenius and Schur, around 1900, and it arrives in crystallography with the character-table methods that a spectroscopist now uses daily. The eighteen is a by-product of that separation rather than a discovery in its own right, and the reason to compute it here is to see how much the separation costs.

Where the ladder goes next #

Forgetting the matrices and keeping the table is one way to weaken a classification. The opposite move — keeping everything and asking when a group contains a copy of itself — is a question about space groups rather than point groups, and its answer turns out to be about which integers a quadratic form represents.

What survives the merge, and what does not #

The collapse is a loss of information, and it is worth being precise about which questions the surviving structure still answers, because the division is sharp.

The multiplication table fixes the representation theory. How many irreducible representations a group has, what their dimensions are, and therefore what degeneracies its levels may carry — all of that is determined by the abstract group. So 222 and mm2 have the same number of levels of each degeneracy, and a spectroscopy counting levels cannot tell them apart.

It fixes nothing about direction. A property tensor is computed by averaging over the group’s matrices, and two classes with one multiplication table have different matrices. That is the whole difference between them, and it is where every physical distinction lives.

The worked case is the same pair. Both 222 and mm2 have four elements, all of order two, with identical tables. And mm2 is polar — it permits a spontaneous electric dipole along its axis, so a crystal in that class can be pyroelectric and ferroelectric — while 222 permits no dipole in any direction at all. One of them can be poled and the other cannot, and their abstract structures are indistinguishable.

The trio at order two makes the point more sharply still. The classes 1̄, 2 and m are one abstract group. The first has a centre, so it permits no property described by an odd-rank tensor: no dipole, no piezoelectricity, no optical activity. The other two permit all three. A single multiplication table therefore covers both the most permissive and the least permissive small class in the subject.

So the eighteen are the right answer to a question, and it is not crystallography’s usual question. They say which groups occur; the thirty-two say which groups occur acting on space, and every prediction about a material comes from the second.