An ideal across and a prime along

The primes a cell can grow by settles which copies of a plane group inside itself are maximal. The criterion is that no lattice the point group preserves lies strictly between the copy’s lattice and the whole; for p4 the invariant lattices are the ideals of ℤ[i], a lattice between two ideals is an ideal dividing one and divided by the other, and a maximal one is a prime ideal. So p4’s maximal indices are the norms of the Gaussian primes — 2, the primes one more than a multiple of four twice over, and the squares of the primes three more — and the hexagonal groups do the same in ℤ[ω].

It ends on a question it cannot answer:

A copy can then grow across the axis by a prime ideal, along it by a prime, or both at once, and the screw axes forbid some of the combinations. Whether the maximal copies of P4₁ are exactly the products of p4’s prime ideals with the primes the screw congruence allows, or whether the two constraints interact, is a question the plane’s answer suggests and does not decide.

They interact, at exactly one kind of index, and the interaction is the enantiomorphic partner.

Two directions, and a table that is a product #

A space group with a single principal axis has a lattice that splits. The cross-section, perpendicular to the axis, is a module over ℤ[i] for a four-fold axis and over ℤ[ω] for a three- or six-fold one, because those are the rings the rotation generates; the direction along the axis is a module over ℤ alone, since nothing in the point group mixes it with the cross-section.

So a sublattice the point group preserves is a choice of ideal across and a choice of multiple along, and its index is the norm of the ideal times the multiple. Those two choices are independent, and the lattice of invariant sublattices is the product of two divisibility orders: one ideal divides another, one whole number divides another, and a sublattice contains a second exactly when both parts do.

A maximal element of a product of two orders has exactly one prime step in it. Take a lattice with a non-trivial step in both directions and the lattice with only the across step lies strictly between it and the whole. So a maximal invariant sublattice is either a prime ideal across with the axis unchanged, or a prime along with the cross-section unchanged, and never both. The plane’s question — whether the maximal indices are products — has a negative answer before any group is mentioned: they are a union rather than a product.

That much is arithmetic about lattices. What a group does on them is the other half.

The congruence that decides the hand #

A screw that contains its own mirror image works out what happens along the axis. A screw of order n advancing q/n of the cell edge per turn, put on a cell p times longer, advances q/(np) in the new edge — which is a legal screw only when p is prime to n, and then the new screw part is q times the inverse of p modulo n.

For P4₁ the inverse of k modulo four is 1 when k ≡ 1 and 3 when k ≡ 3. So the cell three times longer carries P4₃ and the cell five times longer carries P4₁ again, and even cells carry no four-fold screw at all. That is the screw congruence, restated as a table.

Across the axis nothing of the kind happens. An ideal of ℤ[i] changes the cross-section and leaves the translation along the axis exactly where it was, so the screw part does not move. Every invariant sublattice with a trivial along-axis part carries the parent’s own type, and the ideal arithmetic transfers from the plane unchanged.

Maximal across, maximal along, and never both #

Putting the two together gives the row.

P4₁’s maximal copies of itself sit at 2, 5, 9, 13, 17 and 29 up to thirty. At 2 the step is across — the ideal generated by 1 + i, whose norm is two — and along the axis there is nothing, since two is not prime to four. At 5, 13, 17 and 29 there are three copies each: two across, one on each of the two conjugate prime ideals above the split prime, and one along the axis, since those primes are one more than a multiple of four and the cell that many times longer carries P4₁ again. At 9 there is one, across, on the ideal generated by 3.

The two lists are added rather than multiplied, and where they overlap the index carries copies of both kinds. That is the first half of the plane’s question answered, and it is the half the arithmetic of a product order settles on its own.

The index the plane calls maximal and space does not #

The second half is where the two constraints meet, and it is one index.

Take three, which stays prime in ℤ[i]. In the plane p4 has a maximal copy at index nine because the only invariant lattice between the copy’s and the whole would have index three, and no ideal of ℤ[i] has norm three — the prime does not split, so there is nothing to put in between.

Along the axis the lattice of index three does exist. It is the cell three times longer, it is invariant under the four-fold rotation, and a group sits on it. What sits on it is P4₃.

P4₃ is not P4₁, so the copy at index three is not an isomorphic subgroup. But it is a subgroup, and maximality asks whether any subgroup lies strictly between — not whether an isomorphic one does. So the copy of P4₁ at index nine along the axis has P4₃ between it and the whole, and is not maximal.

An index that is maximal in the plane stops being maximal in space, and the thing that takes it away is a group of a different type. That is the interaction the plane’s answer could not decide, and it is a single-place failure rather than a general one: everything else about the two directions really does behave as a product.

Only the chiral screws lose anything #

The obvious next question is which groups this happens to, and the answer separates the family cleanly.

P4, P3 and P6 have no screw at all, so every step along the axis lands on themselves and nothing is lost. P4₂ has a screw whose mirror image is itself — a half-turn advance is the same either way round — so its along-axis steps also land on itself, and it loses nothing.

P4₁, P3₁ and P6₁ have screws with a hand, and each loses exactly the indices that are squares of primes inert in its own ring. P4₁ loses nine; P6₁ loses twenty-five, since five is two modulo three and stays prime in ℤ[ω]; P3₁ loses four and twenty-five, because a three-fold axis admits even along-axis steps and two is inert in ℤ[ω] as well.

Chirality is what breaks the product. A group whose screw has no mirror image to be confused with behaves exactly as the plane predicted, and one whose screw has a partner does not — and the reason is nothing to do with lattices. It is that the partner is a group, and a group between two groups is what maximality is about.

What the Tables print, and why the series look the way they do #

The International Tables list the maximal subgroups of each space group and, among the ones that keep the point group, the isomorphic ones as infinite series indexed by primes. What the computation above supplies is the reason the axial groups’ series have the shape they do.

A series of index p for every prime p ≡ 1 mod 4, with three subgroups at each, is the statement that such a prime splits in ℤ[i], giving two ideals across the axis, and that the cell p times longer carries the group itself, giving one more along it.

A series of index $p^2$ for every prime p ≡ 3 mod 4, with one subgroup each, is the inert case across the axis — and it is not accompanied by a series along the axis at $p^2$, because that copy is not maximal. A reader who derived the entry from the plane’s answer would predict one too many.

And the entries at 2 exist for P4 and P4₂ and not for P4₁, because the ramified Gaussian prime gives an across step at index two for all of them while the along-axis step at two is legal only when the screw has even order. That asymmetry is invisible in the plane, where there is only one direction to grow in.

The lattice across the axis is the plane’s lattice, and that is not a metaphor #

It is worth being exact about what “an ideal across the axis” means, because the phrase can be read as an analogy and is not one.

A four-fold rotation about c carries the a and b axes to each other and leaves c alone. Restricted to the plane of a and b it is the quarter-turn, and the sublattices of that plane which it carries to itself are precisely the ones a quarter-turn keeps — the ideals of ℤ[i], whose indices are the sums of two squares. Nothing about being in space changes them: the cross-section is the plane, the operation restricted to it is the plane’s operation, and the arithmetic is the plane’s arithmetic.

What space adds is the third direction and the screw. So the correct statement of the relationship is not that the space group’s answer resembles the plane group’s; it is that the space group’s answer contains the plane group’s as one of its two factors, unchanged, together with a second factor the plane has no room for.

That makes the ring’s role concrete in a way the plane’s essay could only claim. Which indices admit a sublattice of the same shape is a question about a quadratic form representing an integer, and here the same form does the same work one factor at a time while the other factor counts multiples of a single vector. The Löschian numbers appear for a three-fold axis and the sums of two squares for a four-fold one, exactly as they do in the plane, and the along-axis factor contributes every whole number the congruence allows.

Every copy is a chain, and the chains have two kinds of step #

Maximality is worth a sentence about what it is for. The Tables print maximal subgroups because every descent passes through them one step at a time, so the maximal ones generate everything.

Here that means every copy of P4₁ inside itself is reached by a sequence of prime steps, each across the axis or along it, and the order of the steps does not matter because the two directions are independent. A copy at index forty-five, say, is reached by an across step at nine and an along step at five, or the other way round, and they are the same subgroup.

The chains that pass through the enantiomorph are the interesting ones. A copy of P4₁ at index nine along the axis is reached by two steps of three, and the group in the middle is P4₃ — so a chain of isomorphic subgroups is not the same thing as a chain of subgroups, and the second is what maximality is defined against. Going down two steps returns to the group it started from, which is the behaviour the descent of symmetry shows in general: the routes between two groups form a lattice rather than a chain, and a route may leave the type it started in and come back.

Where the exactness stops #

Computed here. For each of seven axial groups: every invariant sublattice to index thirty as an ideal across and a multiple along, the ideals of each norm in ℤ[i] and ℤ[ω] up to units, the screw part the longer cell carries, which lattices are maximal, and which maximal lattices carry the parent’s own type. Four checks that can fail, including that a group with no screw loses nothing.

The splitting of the lattice is an assumption, not a derivation. That a sublattice invariant under a group with a single principal axis factors as an ideal across times a multiple along is stated and used; it follows from the rotation acting trivially on the axis direction and faithfully on the cross-section, and that argument is not written out. It fails for the cubic groups, where the rotations mix all three directions, and that is a separate calculation.

Nothing here builds a group. The plane’s version of this work built each candidate subgroup by choosing cosets, closing, and handing the result to an identifier. Here the group on a lattice is named by the congruence rather than constructed, which is a shorter argument and a weaker check. The congruence itself was built and checked where a screw contains its own mirror image.

The seven groups are the ones with a single axis and a primitive lattice. Centring adds invariant sublattices of its own — the body-centred tetragonal lattice contains the primitive one at index two and is contained in it — and every count above is on a primitive cell. Which of the centred axial groups gain indices, and which lose them, is the same computation with a larger starting lattice and is not run here.

And the copies are counted as lattices. One lattice may carry more than one copy, and the numbers above count lattices — the same understatement the plane’s answer records and the count of copies on one lattice measures.

The refusal is the plane’s own prediction: that the maximal copies are every product of a prime ideal with an allowed prime. It is nearly right, it is wrong at one index for each chiral group, and the index it is wrong at is the one where a different group turns out to be sitting in the way.

Who worked out the series #

Yves Billiet and Erwin Bertaut tabulated the maximal isomorphic subgroups of the space groups in the 1970s, and their infinite series are what the subgroup volume of the International Tables prints. The arithmetic behind the plane’s half — which primes split, ramify or stay prime in ℤ[i] and ℤ[ω] — is Gauss’s and Eisenstein’s, and was settled by the middle of the nineteenth century.

The interaction on this page does not seem to be anywhere as a statement, and there is a reason it would not be. A table of maximal subgroups is compiled group by group: the entries for P4₁ are computed, checked and printed, and nobody needs to know that the entry at nine which the plane would have predicted is absent, because nobody wrote it down to be absent. A missing entry is invisible unless something predicts it, and what predicts it here is the plane’s own answer taken one dimension up.

Still open: what a second axis does #

Everything above has one axis. A group with two independent axes — the orthorhombic groups with three two-fold axes, or the cubic groups — has a lattice that does not split into a cross-section and a direction, because the rotations mix them.

For the orthorhombic groups the lattice splits three ways rather than two, into three directions each a module over ℤ, so the invariant sublattices are triples of whole numbers and the maximal ones have one prime step in one of the three. The screw congruence then applies in each direction separately, with order two, so an even step in a direction carrying a 2₁ axis is illegal and every odd step keeps the group. That is the same analysis with three factors and no chirality in it, and it is not carried out here.

For the cubic groups nothing splits, the invariant sublattices are very few, and the whole question collapses — which is what the cubic groups’ own arithmetic shows, and which is the opposite extreme from the plane.