A bigger cell, and sometimes the mirror

Almost every subgroup gives something up. Take away an operation and the pattern loses a symmetry; take away a translation and the cell grows and the class survives. Both are descents in the ordinary sense, and both are catalogued.

An isomorphic subgroup gives up nothing but scale. It is the same group again — same symbol, same operations, same everything — on a coarser lattice, and it exists because a lattice contains sublattices of every index. In the plane the available indices turn out to be the values a quadratic form takes.

In space the screw axes make the question sharper, and the answer is not what it looks like.

4_1: which index gives which group. The isomorphic subgroups of a 4₍1₎ screw group, index by index. An index sharing a factor with 4 gives nothing — the translation cannot be written on the new cell at all — and the rest give a screw whose index is the old one times the inverse of p modulo the axis order. So the answer alternates: some indices give the group back and others give its mirror image, and which is which is decided by p modulo the order of the axis. — Fig. 1 The isomorphic subgroups of a four-fold screw group, index by index. Even indices are refused; the odd ones alternate between giving the group back and giving something else, and the something else is the point of this essay.

The arithmetic

A screw axis n_q advances by q/n of the cell edge for every turn of 2π/n. That fraction is the part of the translation no origin removes, and it is what the subscript in the symbol records.

Now measure the same operation against a cell p times taller along the axis. The advance was q/n of the old edge; the old edge is 1/p of the new one; so the advance is q/(np) of the new edge — which is a legal screw only if it can be written as k/n for some whole number k, modulo a whole cell.

The operation may also be composed with any lattice translation before being measured, which adds any whole number m of old cells. So the condition is that

\frac{q + mn}{p} \equiv k \pmod{n}

has a solution, and it does exactly when p is prime to n. Then

k \equiv q\,p^{-1} \pmod{n},

where p⁻¹ is the inverse of p modulo the order of the axis.

So which group the subgroup is depends on p modulo n, and for most values of p it is not the group one started with.

The surprise

Take P4₁, whose screw advances a quarter of the cell per quarter turn. The inverse of p modulo 4 is 1 when p ≡ 1 and 3 when p ≡ 3. So:

a cell five times taller holds P4₁ again, at index five;
a cell three times taller holds P4₃ — the enantiomorphic partner, the screw of the opposite hand — at index three.

A left-handed screw contains a right-handed one, and nothing was done to the crystal. No atom moved; no operation was discarded except the translations the coarser lattice does not have; and the group that remains is the mirror image of the group that was there.

P4_1 ⊃ P4_3 at index 3. The same helix of positions twice, with two different cell rulings drawn across it. On the left the rulings are the parent's cell and the screw advances 1/4 of it per step; on the right they are 3 times further apart and the same positions advance 3/4 of the taller cell. Nothing has moved — what changed is the unit the climb is measured in, and for this index that makes the group its enantiomorphic partner, a screw of the opposite hand obtained with nothing done to the crystal but a change of description. The subgroup was built from the parent's own operations and compared with the named group's, up to an origin shift. — Fig. 2 One helix of positions, measured against two cells. In the group’s own cell the screw advances a quarter per quarter turn. In a cell three times taller the same positions advance three quarters per quarter turn, which is the other hand’s screw. Nothing has moved: what changed is the unit the climb is measured in.

The same happens on every screw whose partner exists. 3₁ contains 3₂ at index two and five, and gives itself back at four and seven. 6₁ contains 6₅ at index five. The pattern is that p and its inverse modulo n are the same when p² ≡ 1, and otherwise they are not.

3_1: which index gives which group. The isomorphic subgroups of a 3₍1₎ screw group, index by index. An index sharing a factor with 3 gives nothing — the translation cannot be written on the new cell at all — and the rest give a screw whose index is the old one times the inverse of p modulo the axis order. So the answer alternates: some indices give the group back and others give its mirror image, and which is which is decided by p modulo the order of the axis. — Fig. 3 The three-fold screw’s series. Indices divisible by three are refused; the rest alternate, with two, five, eight and eleven giving the partner and four, seven, ten and thirteen giving the group back. The alternation has period three because the inverse is taken modulo three, and it never settles down.

Why the even indices are refused

A cell twice as tall cannot hold a four-fold screw at all, and the reason is worth doing explicitly because it is the same arithmetic in a different mood.

The operation’s advance is a quarter of the old cell, which is an eighth of the new one. Composing with lattice translations gives an eighth, three eighths, five eighths, seven eighths — never a quarter, never a half, never nothing. An eighth is not k/4 for any whole k, so there is no legal screw, and the subgroup on that lattice is not the group.

Something does survive: the group generated by the four-fold’s square, which is a two-fold screw and does fit. But that is a smaller group, not an isomorphic one, and it is a different kind of descent.

The general condition is gcd(p, n) = 1, and it says that a lattice may only be coarsened along a screw axis by a factor sharing no divisor with the order of the axis.

The table it acts on is small enough to hold in mind. There are eleven screw axes — 2₁; 3₁ and 3₂; 4₁, 4₂ and 4₃; 6₁ through 6₅ — one for each non-zero subscript at each of the four rotation orders a lattice permits, and each advances q/n of the cell per turn. Eight of the eleven have a partner obtained by replacing q with n − q, and the other three are 2₁, 4₂ and 6₃, whose advance is exactly half a cell and which are therefore their own partner. The arithmetic above never crosses for those three: run the series on 2₁ and every available index returns 2₁, because the inverse of an odd number modulo two is one and there is no other subscript for it to reach.

6_1: which index gives which group. The isomorphic subgroups of a 6₍1₎ screw group, index by index. An index sharing a factor with 6 gives nothing — the translation cannot be written on the new cell at all — and the rest give a screw whose index is the old one times the inverse of p modulo the axis order. So the answer alternates: some indices give the group back and others give its mirror image, and which is which is decided by p modulo the order of the axis. — Fig. 4 The six-fold screw’s series, where both kinds of refusal happen at once. Six shares a factor with every even index and with every multiple of three, so only four of the twelve indices up to thirteen are available at all — five, seven, eleven and thirteen — and of those, five and eleven give 6₅ while seven and thirteen give 6₁ back. A six-fold axis is the most constrained case in the table and the one where the condition is hardest to mistake for something about parity.

The arithmetic against the construction

One line of arithmetic is not enough to publish, so every row of the table was also built.

The parent’s generator is taken with its own translation, the translation is measured against the taller cell in each of the p ways it could have been measured, and the closure of each candidate is formed. Exactly one of the p choices generates a group of the right order. The others pick up an extra translation — a third of the new cell, say — which is a translation the parent has and the subgroup must not, and the closure runs away to a larger group.

The survivor is then compared with each candidate group’s own operations, up to a choice of origin, which is the only freedom a space group has left.

24 of 24 agree. One line of arithmetic against one construction. The arithmetic says which screw a cell p times taller carries; the construction takes the parent's operations, keeps the ones the new lattice admits, and compares what is left with each candidate group up to a choice of origin. Every row agrees, including the rows where the answer is the enantiomorphic partner rather than the group itself — which is the row a reader would want to check. — Fig. 5 Twenty-four rows: two four-fold groups and two three-fold ones, at every index up to seven, with what the arithmetic predicts beside what the construction produced. They agree everywhere, including on the rows where the answer is the partner rather than the group — which are the rows a reader would want to check.

The construction is not a restatement of the arithmetic. Its acceptance criterion is that the generated group has the right order, which is a fact about closure rather than about modular inverses, and its identification is a set comparison against a group built independently. A wrong formula would produce a mismatch rather than a silent agreement.

What an isomorphic subgroup looks like from inside

It is worth being concrete about what the subgroup is, since “the same group on a coarser lattice” is a description and not a construction.

Start with all the operations of P4₁: every four-fold screw about every axis position, every two-fold that is its square, and every lattice translation. Now discard the translations that are not multiples of three cells along c, and discard every operation whose translation part is not what remains. What is left is still closed — a product of two survivors survives — and it is still infinite, and it still has a four-fold screw in it.

The screw it has is the one whose advance is three quarters of the new cell. That is not a different operation from the parent’s screw; it is the parent’s screw, composed with two lattice translations, measured in a different unit. All three descriptions name the same motion of space.

So the subgroup contains a third of the parent’s translations and all of its point operations, which is why the index is three. And its symbol is P4₃ because the symbol is read off the operations in the group’s own basis, which is now the coarse one.

2 screw axes. 2 of the eleven screw axes a lattice permits, each drawn as the helix it is: 4₁, 4₃. A turn of 2π/n followed by an advance of m/n of the repeat, so that n turns land exactly m cells along. 0 of those drawn are its own mirror image; the rest come in left- and right-handed pairs. — Fig. 6 The two four-fold screws drawn as the orbits they are: a quarter turn with a quarter climb, and a quarter turn with three quarters. One is the other seen through a cell three times taller, and the drawings differ only in how much of the vertical scale is shown — which is the entire content of the enantiomorphic relation.

What this says about the enantiomorphic pair

The eleven enantiomorphic pairs are usually introduced as the reason two hundred and thirty and two hundred and nineteen are both correct: the two members of a pair are the same group up to an orientation-reversing change of coordinates and different groups up to a proper one.

The subgroup relation says something stronger and stranger. The two members of a pair each contain the other, at infinitely many indices, as isomorphic subgroups. P4₁ contains P4₃ at index three, seven, eleven; P4₃ contains P4₁ at the same indices; and neither containment involves any reflection anywhere.

That is not a contradiction. A group can contain a subgroup isomorphic to a group it is not isomorphic to — the integers contain the even integers, which are isomorphic to the integers — and here the containment is proper and infinite-index-free in exactly that way. The hand of a screw is a property of the group together with its lattice, and coarsening the lattice is enough to change it.

P4₁, in the two diagrams the Tables print. Space group P4₁, number 76, projected down c on a primitive tetragonal cell. The symmetry elements drawn: 4 4₁ screw axes, 2 2₁ screw axes. 4 general positions, the orbit of one point, each labelled with its height along c and marked with a comma where the operation that produced it reversed handedness. — Fig. 7 P4₁ in plan and axonometric: a four-fold screw climbing a quarter of the cell per quarter turn, and the orbit of one position under it. Reading the same orbit against a cell three times taller gives P4₃’s plan exactly — same points, same axis, a different unit on the vertical scale.

Where the reader should be careful

Three cautions, and the first is about what “the same group” is being claimed.

Isomorphic, not equal. The subgroup is a different subset of the same motions, isomorphic to the parent as an abstract group and equal to it in type. The parent has translations the subgroup does not.

A different setting is not a different group. Half the confusion in this corner of the subject is between a group and its description. P4₁ measured against a taller cell is P4₃ — not “looks like”, not “is equivalent to”. The operations, written in the new basis, are P4₃’s operations. What changed is which lattice is called the lattice.

And this is a statement about groups, not about crystals. A real crystal in P4₁ does not “contain” a P4₃ crystal in any physical sense. What it contains is a subgroup of its symmetry group, and the subgroup describes the same atoms with fewer translations counted. Nothing about the material changes, which is exactly what makes the arithmetic so easy to over-read.

The plane makes the same move and never gets the surprise, which is the cleanest way to see where the surprise comes from. Tripling a plane cell along one axis keeps a third of the translations and all of the point operations, exactly as here, and the group that survives is the group one started with whenever the coarsening is one the point group preserves. Nothing crosses to anything, because there is nothing to cross to. The extra ingredient in space is not the extra dimension and not the larger catalogue: it is that an operation may carry a translation of its own, and a translation of its own is measured in cells, so changing the cell changes the number that names it.

Where this is used

Two places, and both are more practical than the arithmetic suggests.

Superstructures. An ordering transition that multiplies the cell along one axis produces exactly this situation: the child group is an isomorphic subgroup of the parent, the extra reflections are the superlattice reflections, and knowing which index is possible tells a crystallographer which multiplications to test for.

Twinning and domain counting. The index of an isomorphic subgroup is the number of translational domain states the transition produces, by the same coset argument that counts every other kind of domain. An index-three subgroup means three antiphase domains, and their boundaries are where the ordering is out of step.

P3_1 ⊃ P3_2 at index 2. The same helix of positions twice, with two different cell rulings drawn across it. On the left the rulings are the parent's cell and the screw advances 1/3 of it per step; on the right they are 2 times further apart and the same positions advance 2/3 of the taller cell. Nothing has moved — what changed is the unit the climb is measured in, and for this index that makes the group its enantiomorphic partner, a screw of the opposite hand obtained with nothing done to the crystal but a change of description. The subgroup was built from the parent's own operations and compared with the named group's, up to an origin shift. — Fig. 8 The three-fold case at index two: P3₁ measured against a doubled cell, which is P3₂. Two is the smallest index at which a screw group crosses to its partner, and the crossing is visible in the drawing as a helix whose apparent pitch has changed while its positions have not.

The plane’s version, for comparison

The plane has no screws, so nothing there crosses to a partner — but the same question has an answer worth putting beside this one.

In the plane an isomorphic subgroup of index n exists when the lattice has a sublattice of index n preserved by the point group, and which n those are is a question about which integers a quadratic form represents. For p4 the form is a² + b², so the available indices are the sums of two squares: 1, 2, 4, 5, 8, 9, 10, 13. For p3 and p6 the form is a² + ab + b², and the indices are the Loeschian numbers.

The three-dimensional version has the same shape and one more ingredient. The sublattice question is still there — which coarsenings the point group preserves — and on top of it sits the screw condition, which is about the translation attached to an operation rather than about the lattice. The plane has only the first; a group with no screw has only the first in space too; and the crossing between enantiomorphs comes entirely from the second.

So the plane’s spectrum is a list of available indices and nothing more: for p4 the answers are 1, 2, 4, 5, 8, 9, 10, 13 and every one of them gives p4 back. There is no second column to fill in, because the plane’s five groups that a single hand may sit in are each their own mirror image and none of them has a partner to be confused with. The second column exists only where an operation has a subscript.

One more index worth doing by hand

Index seven on P4₁, because the number is large enough that the pattern is not a coincidence.

Seven is prime to four, so the subgroup exists. The inverse of seven modulo four is three, since 7 × 3 = 21 = 1 modulo 4. So k = 1 × 3 = 3, and the subgroup is P4₃.

Checking it directly: the parent’s screw advances a quarter of the old cell. Measured against a cell seven times taller, that is a twenty-eighth; composing with m whole old cells gives (1 + 4m)/28 of the new cell, and the values of that for m = 0, 1, 2, … are 1/28, 5/28, 9/28, 13/28, 17/28, 21/28. The last of those is 3/4 — a legal four-fold screw, of the opposite hand — and it is the only one of the seven that is k/4 for a whole k.

P4_3 ⊃ P4_1 at index 3. The same helix of positions twice, with two different cell rulings drawn across it. On the left the rulings are the parent's cell and the screw advances 3/4 of it per step; on the right they are 3 times further apart and the same positions advance 1/4 of the taller cell. Nothing has moved — what changed is the unit the climb is measured in, and for this index that makes the group its enantiomorphic partner, a screw of the opposite hand obtained with nothing done to the crystal but a change of description. The subgroup was built from the parent's own operations and compared with the named group's, up to an origin shift. — Fig. 9 The relation run the other way: P4₃ against a cell three times taller, which is P4₁. Each member of the pair contains the other at the same indices, and neither containment involves a reflection anywhere — so the two groups are related both by a mirror and by a change of cell, which are entirely different operations arriving at the same place.

Six candidates refused and one accepted, with the accepted one being the partner. That is the construction and the arithmetic agreeing on a case small enough to write out, which is the level of checking a claim of this kind deserves before the table is believed.

Who worked it out

The isomorphic subgroups were catalogued as part of the International Tables’ programme of listing every maximal subgroup of every space group, which took most of the twentieth century and was completed in the volume on symmetry relations published in 2004. Billiet and Bertaut worked out the general theory of the isomorphic case in the 1970s; the observation that the series crosses between enantiomorphs is in their treatment and is usually stated as a table rather than as an argument.

The arithmetic is elementary and the tables are enormous, which is the usual shape of this subject: the general rule fits in a line and the two hundred and thirty instances of it fill a book.

The series never stops, and that matters

A finite group has finitely many subgroups. A space group has infinitely many, and the isomorphic ones are why: for every index prime to the axis order there is one, and the primes go on for ever.

That has a consequence for how the subgroup structure can be catalogued at all. The maximal subgroups can be listed — Hermann’s theorem says a maximal subgroup gives up operations or translations and never both — but the isomorphic ones form infinite series, so the Tables list them as series with a parameter rather than one by one: “P4₁, index p, p prime to 4, giving P4₁ for p ≡ 1 and P4₃ for p ≡ 3 modulo 4”, which is this essay’s arithmetic in the form a table can hold.

A classification with an infinite family in it is not a failed classification, and this is the standard example. What is finite is the number of shapes the family takes, and here that number is two.

Which partner, from a group acting on the subscripts

The formula k ≡ q p⁻¹ (mod n) says which screw a given index produces, and it is worth reading as an action rather than as a lookup, because doing so answers a question the table only illustrates.

The indices p that give a subgroup at all are those prime to n — which is to say the units modulo n — and each of them sends the subscript q to q p⁻¹. That is a group acting on the set of subscripts, and which screws are reachable from which is the orbit structure of that action.

Run it. For a four-fold axis the units modulo four are 1 and 3, so the orbit of q = 1 is {1, 3}: P4₁ and P4₃ are one orbit and reach each other, while q = 2 is fixed, because 2 × 3 = 6 ≡ 2. For a three-fold axis the units are 1 and 2, so 3₁ and 3₂ are one orbit. For a six-fold axis the units are 1 and 5, giving the orbits {1, 5}, {2, 4} and the fixed point {3}.

Those orbits are the enantiomorphic pairs, and their fixed points are exactly 2₁, 4₂ and 6₃ — the three screws that are their own mirror image. That is not a second fact needing its own argument: p ≡ −1 is always a unit, it sends q to −q ≡ n − q, and a subscript fixed by it is one with 2q ≡ 0.

So the essay’s surprise is the action’s orbit of size two, and the reason some screws have no partner is that they are its fixed points. One modular inverse explains both.

11 screws, 4 partner pairs, 3 on the middle. Every screw axis a lattice permits, placed at its own pitch — the fraction of the cell it climbs per turn — with a bracket joining the ones that are each other's enantiomorphic partner. The indices at which an isomorphic subgroup exists are the units modulo the axis order, each of them sends a subscript q to q times its inverse, and the screws reachable from one another are that action's orbits. Those orbits are computed here and compared with the orbits of q ↦ n − q, the reflection about half a cell: they agree at every permitted order, so a partner pair sits symmetrically about the middle of the scale and the 3 screws that are their own partner sit exactly on it. The last row is why that agreement is not general. Modulo eight the units are 1, 3, 5 and 7 rather than ±1, an orbit has 4 members instead of two, and an eight-fold screw would have three partners — so screws pairing off is a consequence of the crystallographic restriction rather than a fact about screws. — Fig. 10 The eleven screw axes placed at their own pitches — the fraction of a cell each climbs per turn — with a bracket joining the ones that are each other’s partner. The orbits drawn are the orbits of the unit action, computed from the modular inverses, and they are compared against the orbits of q ↦ n − q before the figure is returned. Because those agree, a pair sits symmetrically about half a cell and the three self-partnered screws sit exactly on it. The last row is the eight-fold case, which no lattice permits: there the units are 1, 3, 5 and 7 rather than ±1, the orbit has four members, and a screw would have three partners.

And that last row is the part worth keeping. The agreement between the two routes is not a general fact about screws; it holds because the unit group modulo each of 2, 3, 4 and 6 happens to be {±1}, and there is nothing else for a unit to be. Modulo eight there is: the units are 1, 3, 5 and 7, the orbit of a subscript has four members rather than two, and an eight-fold screw would sit in a set of four mutually reachable screws instead of a pair. So enantiomorphic screws coming in pairs is another consequence of the crystallographic restriction, arriving from a direction that has nothing to do with traces of integer matrices.

The eleven pairs, counted from the same action

That reading gives the enantiomorphic pairs a derivation rather than a list, and it is worth completing because the number eleven appears throughout this collection without one.

A pair exists wherever a screw’s subscript is not fixed by q ↦ n − q, which happens for n = 3 at q = 1, 2; for n = 4 at q = 1, 3; and for n = 6 at q = 1, 2, 4, 5. Each such orbit of size two contributes one pair of screw types, and a pair of space groups arises for each way those screws sit in a group with the rest of its operations.

Counting those arrangements over the trigonal, tetragonal and hexagonal systems gives eleven, which is the number the enantiomorph essay reports and which is otherwise quoted from the tables. The two accounts have to agree, and the point of having both is that one of them says why the number is what it is: it counts the non-fixed orbits of an action of the units modulo the axis order.

And the same action explains why no pair is built on a two-fold screw. The units modulo two are just the identity, so every subscript is fixed and there is nothing to pair — which is the arithmetic form of the observation that 2₁ and its mirror image are the same operation.

Where the ladder goes next

Downwards, into what a subgroup relation is for. The descent of symmetry is a lattice, not a tree is the general structure — a group has many maximal subgroups and the routes between two groups form a lattice rather than a chain — and the isomorphic subgroups are the part of that structure that never terminates, since every group has them at infinitely many indices.

Sideways, into the other thing a change of cell can do. One group, three symbols is the case where the group does not change at all and only its name does, which is the opposite failure mode and considerably commoner in the literature.