How many axes there are is a Sylow count

A crystal class is a finite group, and finite groups have theorems. Sylow’s are the oldest useful ones and they say three things: for every prime p dividing the order there is a subgroup whose order is the largest power of p dividing it; all such subgroups are conjugate; and how many there are is congruent to one modulo p and divides the rest of the order.

None of that mentions an axis. Applied to the thirty-two it counts them.

The three theorems are worth stating once in the form used here, because the third does the work and it is the one usually skipped. Let |G| = p^e · m with p not dividing m. Then G has a subgroup of order p^e; any two such are conjugate; and their number n satisfies n ≡ 1 (mod p) and n | m. The first is an existence statement, the second says they cannot be told apart, and the third is a numerical constraint tight enough to settle many cases on its own.

Conjugate subgroups are equivalent directions #

A Sylow 3-subgroup of a crystal class is generated by a rotation of order three, so it belongs to a direction. Two such subgroups are conjugate exactly when some operation of the class carries one direction onto the other — which is what “symmetry-equivalent” means. So:

the number of Sylow 3-subgroups is the number of symmetry-equivalent threefold axis directions.

That is a translation, not a theorem, and its use is that Sylow’s third statement constrains the number before any axis is looked at.

One caution before the table. A Sylow subgroup is a subgroup of maximal prime-power order, not any subgroup of prime order, and the distinction matters at the prime two. In a class of order twenty-four the Sylow 2-subgroup has order eight, not two, so it is not a single axis and its conjugates are not single directions. At the prime three the largest power dividing any crystal class’s order is three itself — no crystal class has order divisible by nine — so a Sylow 3-subgroup always is a single threefold rotation and its two powers, which is why the threefold reading is exact and the twofold one is not.

That last fact deserves a sentence of its own, because it is what makes the whole essay work and it is a small crystallographic miracle. The orders of the thirty-two are 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48, and not one of them is divisible by nine. So the prime three enters every class at the first power only, and the Sylow theory at that prime is as simple as it can be.

Three values in a table of forty-seven entries is the first thing worth noticing, and it is Sylow’s third statement doing the work rather than any property of crystals. At these orders the numbers congruent to one modulo two that divide the index are one and three; the numbers congruent to one modulo three are one and four. There is very little room.

It is worth spelling out why conjugacy is the right notion and equality is not. Two threefold rotations about different axes generate different subgroups, and asking whether those subgroups are equal is a question about the axes being the same line. Asking whether they are conjugate is a question about the class containing an operation that carries one line to the other — which is the question a crystallographer means by “are these two axes equivalent”. So the group-theoretic relation and the crystallographic one are the same relation, and it is Sylow’s second statement that says all the Sylow subgroups at a given prime are related by it.

That has a consequence worth stating separately. Because all Sylow p-subgroups are conjugate, a class cannot have two inequivalent families of threefold axes: whatever threefold directions it has, they are all equivalent to each other. That is a strong structural fact and it is not true of other orders — a class can perfectly well have two inequivalent kinds of twofold axis, and the hexagonal classes do — because a subgroup of order two need not be a Sylow subgroup.

What “cubic” is, without a cube #

That is the result of the essay. A crystal class is cubic exactly when its Sylow 3-subgroup is not normal, and the statement contains no lattice, no cell and no axis — only a group and a prime.

It also explains the thing about cubic classes that a beginner finds strange. Every other system has a principal axis, and the cubic system does not; the standard explanation is that it has four threefold axes and none of them is special. The Sylow reading says why none of them can be: they are conjugate, and a group has no way to distinguish members of a conjugate family. Singling one out would mean choosing a coset representative, which is a description rather than a property.

The reason the count is four rather than some other admissible value is worth chasing, since the theorem allows one or four and nothing else. Four is the index of the normaliser of a threefold subgroup, and in a cubic class that normaliser has order six or twelve against a group of twenty-four or forty-eight. In geometric terms: the operations fixing one body diagonal are the rotations about it and whatever mirrors contain it, and there are exactly a quarter as many of those as there are operations in all. So four is the orbit-stabiliser theorem applied to a set of axes, and Sylow’s contribution is the guarantee that the set is a single orbit.

There is a converse reading which is the one a crystallographer will find useful. A class with a unique threefold direction has a normal Sylow 3-subgroup, and a normal subgroup gives a quotient — so every trigonal and hexagonal class maps onto a group of order |G|/3 by forgetting the threefold rotation. That quotient is what a description “a threefold axis with such-and-such attached to it” means, made precise. A cubic class has no such quotient at the prime three, which is why cubic symbols are not built as “an axis plus decorations” the way every other system’s are.

A reader who prefers the geometric statement can have it back at once, and the exchange rate is worth seeing. “Four equivalent threefold axes” and “four conjugate subgroups of order three” are the same sentence; “no principal axis” and “no normal Sylow 3-subgroup” are the same sentence; “the threefold axes of a cubic class are all alike” is Sylow’s second statement and nothing else. What the group-theoretic form buys is the third statement, which has no geometric counterpart at all: a constraint on how many there can be, obtained by dividing.

The theorem often settles it before the group is looked at #

The pair 432 and 6/mmm is the case to read. Both have order twenty-four; both have an index of eight at the prime three, so both are allowed one or four Sylow 3-subgroups. 6/mmm takes one and 432 takes four, and that difference is the difference between a class with a principal axis and a class without one. The theorem narrows the possibilities to two and the group decides between them — which is the usual division of labour between an arithmetic constraint and an object.

The 23 and m3̅ rows deserve a note, because they are the cubic classes without fourfold axes and they behave differently at the prime two: their Sylow 2-subgroups are normal, so only the threefold family is conjugate. So the cubic system is not uniformly “everything is conjugate to everything” — 23 has a normal subgroup of order four and 432 does not, and that difference is what makes 23 solvable in a way 432 is not. The prime-by-prime reading separates classes that a system name lumps together.

There is a broader point in that pair which is worth extracting. An arithmetic constraint of this kind is cheap — it costs a division and a congruence — and it eliminates possibilities that would otherwise have to be ruled out by construction. This collection has used that shape before: the crystallographic restriction eliminates every rotation order but five by asking whether a trace is an integer, and no lattice is examined. Sylow’s third statement is the same manoeuvre one level up, on subgroups rather than on elements.

The difference is that the restriction settles its question completely and Sylow’s does not. A trace that is not an integer is impossible; a count that is congruent to one modulo three is merely allowed. So the two are complementary rather than parallel: one is a prohibition and the other is a shortlist, and a shortlist of length two is still useful when the alternative is a subgroup enumeration.

The same thing one prime down #

The prime two behaves the same way and says something different, because a Sylow 2-subgroup of a crystal class is usually not a single axis. In 622 it has order four and contains a twofold rotation about a horizontal axis together with the vertical twofold; the three conjugates correspond to the three inequivalent horizontal directions, which is exactly what the symbol’s second position records. So the count reproduces a fact the Hermann–Mauguin symbol already carries, from a completely different place.

The ten classes with a non-normal 2-Sylow are the trigonal and hexagonal classes with horizontal twofolds, together with the three cubic classes containing fourfold or 4̄ operations. That is a list a reader could assemble by looking at symbols; getting it from a prime and a congruence is the point.

One more remark about the ten. Nine of the ten have a principal axis and the tenth family — the cubic ones — do not, so the non-normality at the prime two has two quite different geometric causes: a set of horizontal twofolds exchanged by a vertical rotation, and a set of fourfolds exchanged by a threefold. The count is three in both cases and the mechanism is not the same, which is a reminder that a Sylow count is a number and the interpretation is a separate step.

There is one more entry in the table worth reading aloud. The class 3 itself has order three and is its own Sylow 3-subgroup, so the count is one and the theorem says nothing. The class 32 has order six and a normal 3-Sylow with three conjugate 2-Sylows; 23 has order twelve and reverses that, with a normal 2-Sylow and four conjugate 3-Sylows. Three classes containing threefold rotations, three different Sylow structures, and the structure is what the crystallographic name is recording — trigonal with a unique axis, trigonal with horizontal twofolds, cubic. The names were chosen for other reasons and they line up.

Why this is worth having #

Three reasons, and the third is the general one.

The first is that it removes an appeal to geometry from a definition. “Cubic” is normally defined by the lattice — a class is cubic if it acts on a cubic lattice — and that is circular in the sense that the lattice is classified by the class acting on it. Thirty-two and no others enumerates the classes without a lattice, and this supplies the system boundary without one either.

The second is that it is a check on the enumeration rather than a restatement of it. The Sylow counts are computed from the subgroup lattices of thirty-two matrix groups, and the congruence n ≡ 1 (mod p) is a theorem those counts must satisfy. Forty-seven counts satisfy it. That is a test the enumeration would fail if a group had been closed wrongly or a subgroup missed, and it is a test that no amount of comparing against a published table provides — a published table would agree with a wrong enumeration if the same table was used to build it.

The third is the general shape. A great deal of what this collection computes about the thirty-two — which axes a class pins down, which magnetism a class permits, how many domains a transition makes — is a statement about conjugacy, orbits and normality, dressed in geometry. Once that is noticed, the theorems about finite groups become available, and they are stronger than anything the geometry supplies: Sylow’s third statement is a constraint on a count that no drawing could suggest.

The fourth of those is the one that stops the result being a coincidence of size. 6/mmm has order twenty-four, exactly like 432, and if the non-normality were about the order rather than about the group then the two would behave alike. They do not, and checking it is the difference between “the cubic classes are the large ones” and the statement actually made.

A last observation, and it is the reason the whole exercise is more than a curiosity. Every fact in this essay was obtained from thirty-two matrix groups and two primes, with no reference to lattices, cells, symbols or drawings — and it reproduces the system boundary, the axis multiplicities, and the reason cubic classes have no principal direction. Reading a class off its axes does the same work from the other end, starting from a stereogram and recovering the group. Two routes to the same partition, sharing no step, is the strongest evidence available that the partition is a property of the objects and not of either description.

One consequence for the space groups is worth recording even though it is not computed here. A space group’s point group is one of the thirty-two, so all of this applies to it — and a normal Sylow subgroup of the point group lifts to a question about the space group’s own subgroup structure, which is what a maximal-subgroup chain is made of. The two ways down distinguishes descents that keep the translations from those that change them, and the Sylow structure of the point group constrains the first kind before any translation is considered.

It is worth being explicit that none of this is a new fact about crystals. Every count in the table is visible in a stereogram, and a crystallographer reading a symbol knows how many threefold axes a class has. What is new is the route: the counts are consequences of a theorem about finite groups, they satisfy a congruence that no geometric argument would suggest, and the boundary of the cubic system falls out of a question about normality. A known answer reached by an unrelated argument is worth having precisely because the two arguments can then check each other.

One further use is available and is worth naming because it costs nothing. The Sylow structure is a conjugation invariant, so two classes with different Sylow counts cannot be isomorphic as abstract groups — which settles at a glance several of the pairs that thirty-two classes, eighteen groups is about, where distinct crystal classes turn out to be the same abstract group. Where the counts agree the question stays open and needs the character-theoretic test that essay uses; where they differ, one division ends it.

Where this stops #

Sylow’s theorems say nothing about a group that is already a p-group, and eleven of the thirty-two are: every class whose order is a power of two is its own Sylow 2-subgroup, so the count is one for a reason that carries no information. 4/mmm at order sixteen is the extreme case. So the tool is silent on exactly the classes with the simplest axis structure, which is the right way round but is worth knowing before reading too much into a column of ones.

And the translation from a Sylow subgroup to an axis is one this essay makes and does not prove in general. It is exact when the subgroup is cyclic and generated by a rotation, which is every Sylow 3-subgroup here; it is looser at the prime two, where a Sylow 2-subgroup can have order sixteen and no single axis to belong to. The count there is still the number of conjugates and still means something — it is the size of an orbit of subgroups — but reading it as a number of axes is a shorthand that the 622 case happens to justify and the general case does not.