The same group in a bigger cell

A subgroup of a plane group loses something. The two ways down are losing operations — keeping the lattice and giving up part of the point group — and losing translations, keeping the point group and thinning the lattice.

The second kind can lose nothing at all. Thin the lattice by a factor that the point group tolerates, keep every operation, and the result is the same plane group again, sitting inside itself with a bigger cell. It is a proper subgroup, and there is no way to tell it apart from its parent except by its scale.

The index-five case is the one to look at first, because it is the smallest that is not a simple scaling. Its sublattice is turned relative to the parent, not merely enlarged — the vector (2, 1) and its quarter-turn image (−1, 2) generate it — and the turn is what makes five available where three is not.

The question is at which indices this is possible, and the answer is not “all of them”. A subgroup of any index whatever exists in every plane group — thin the lattice by that factor along one axis and something always comes back — but what comes back is usually a different group, because a sublattice that is not carried onto itself by the parent’s rotations cannot carry the parent’s rotations. So the question is really a question about the lattice rather than about the group: for which n does the parent’s own point group have an invariant sublattice of index n? Once that is asked, the answer stops being about crystallography and becomes about which integers are norms.

The grid #

Asking that question of all seventeen groups at every index up to sixteen gives a grid of two hundred and seventy-two answers, each one a search rather than a lookup: the sublattices of that index are enumerated, those the point group preserves are kept, every assignment of cosets to the parent’s generators is tried, and whatever closes up is handed to an identifier that compares it against all seventeen groups. A filled circle means the identifier returned the parent’s own symbol.

p1, p2, pm, pg, pmm and pmg take every index. Their point groups permit a lattice to be stretched along one direction by any factor, so a sublattice of index n always exists and always has the same kind of metric.

p4 takes 1, 2, 4, 5, 8, 9, 10, 13, 16. Those are the sums of two squares.

p3 and p6 take 1, 3, 4, 7, 9, 12, 13, 16. Those are the Loeschian numbers, the values of a² + ab + b².

And the groups with mirrors take fewer still. p4m takes only 1, 2, 4, 8, 9, 16; p4g takes only the squares; p3m1 and p31m take only the squares; p6m takes 1, 3, 4, 9, 12, 16.

Why a norm #

The pattern in the four-fold row is the arithmetic of the Gaussian integers, and the argument is short.

A sublattice of the square lattice that is carried onto itself by a quarter turn is, in the language of ℤ[i], an ideal — a subset closed under addition and under multiplication by i, which is what a quarter turn is. Every ideal of ℤ[i] is principal, generated by a single Gaussian integer a + bi, and the index of the ideal is the norm a² + b².

So the indices at which p4 contains a copy of itself are exactly the integers that are sums of two squares — 1, 2, 4, 5, 8, 9, 10, 13 — and three, six, seven, eleven and twelve are absent for the reason Fermat gave: a prime congruent to 3 modulo 4 must occur to an even power.

The three-fold case is the same statement in the Eisenstein integers, where multiplication by a primitive sixth root of unity is a sixty-degree rotation, ideals are again principal, and the norm is a² + ab + b².

These are the same two forms whose values are the lengths a lattice has. That is not an analogy: counting sublattices and counting vectors are both questions about which integers are norms in the same ring, and the ring is what the rotation supplies.

Where the phenomenon shows up #

An isomorphic subgroup sounds like an abstraction and is met constantly under other names.

A superstructure. When an alloy orders, its cell becomes a multiple of the parent’s and its space group becomes a subgroup of index equal to that multiple. If the subgroup is isomorphic to the parent, the ordered structure has the same symmetry as the disordered one at a larger scale — and the only sign in a diffraction pattern is the extra reflections the superlattice adds, which appear at fractional positions of the parent’s indices.

A modulated structure locking in. An incommensurate modulation that locks to a rational wavevector produces exactly this: a supercell, the same point group, and a group of finite index inside the parent.

A coincidence-site lattice. Two crystals meeting at a boundary share a sublattice, and its index Σ is constrained by the same arithmetic — for the cubic case, Σ is always odd, which is that system’s version of the norm condition here.

And a colour group. Colouring a pattern in n colours means finding a subgroup of index n, and when the subgroup is isomorphic the coloured pattern has the uncoloured one’s symmetry with the colours permuted. Which is why the groups with short rows in the grid above are the ones that are hard to colour.

What the mirrors take away #

The rows with mirrors are shorter than the rows without, and the reason is worth stating because it is the one place where the arithmetic is not the whole story.

An isomorphic subgroup needs a sublattice invariant under the whole point group, mirrors included. A sublattice of the square lattice invariant under the four-fold rotation may be an ideal generated by any a + bi; requiring it to be invariant under the mirrors as well forces the generator to lie along a mirror or a diagonal, so a = 0 or a = b, and the index is m² or 2m².

That is exactly what p4m’s row shows: 1, 2, 4, 8, 9, 16 are the numbers m² and 2m².

p4g is shorter still, and the culprit is the glide. Its axial lines carry glides, whose translation is half a cell edge; thinning the lattice by a factor that is not a perfect square moves those half-translations to positions the new cell cannot express as glides, and the group that comes back is not p4g. Only the square indices survive — the sublattices that are the parent scaled uniformly.

pgg does the same thing in the rectangular system, keeping only the odd indices, and for the same reason: a glide’s half-translation has to remain half of a lattice vector, and an even scaling along a glide direction destroys that.

What is actually built #

Nothing above is a table. For each index, the procedure is:

Enumerate the sublattices of that index in Hermite normal form — one form per sublattice, no repeats — and keep those the point group preserves.

Reduce the basis. Hermite form is the right way to enumerate and a poor basis to work in: the index-ten sublattice of the square lattice comes out with columns (10, 0) and (3, 1), and its four-fold rotation is the integer matrix [[−3, −1], [10, 3]]. That is a rotation of order four and nothing recognises it as one. Gauss-reducing the pair in the parent’s metric gives short vectors and small matrices. Before that reduction the search reported that p4 had no isomorphic subgroup of index five, ten or thirteen, which is a false negative that looked exactly like a result.

Build the subgroup. For each generator of the parent, choose which coset of the sublattice it sits in, close under composition modulo the sublattice, and require the closure to have exactly one element per point operation.

Try every coset assignment, not the obvious one. For a symmorphic group the zero shift works and the search finds it first; for a group with a glide the obvious choice usually fails and another coset succeeds, so trying only the parent’s own translation parts would report a false impossibility. The number of assignments is the index raised to the number of generators, which is small enough to exhaust at these indices and is the reason the grid stops where it does.

Identify it. Express the result in the sublattice’s own basis and hand it to the plane-group identifier — the same routine the colour groups use, which searches changes of basis and origin shifts against all seventeen and knows nothing about where its input came from. The index counts only if the answer comes back as the parent’s own name.

And the identifier is what makes the grid a result rather than a construction. Seventy-four subgroups of index two are spread across the seventeen plane groups, and most of them come back with a different symbol from their parent’s — that is the ordinary case, and it is why the word “isomorphic” earns its own grid. The entries in the grid above are the minority where the symbol comes back unchanged, and nothing in the construction arranges for that: the subgroup is built from a sublattice and a choice of cosets, handed to a routine that compares it against all seventeen, and the answer is read off. A construction that assumed its own conclusion would fill the grid completely.

A row-by-row reading #

The grid rewards being read slowly, because each row has a reason.

p1 — every index. A lattice with no symmetry to preserve can be thinned any way at all, and σ(n) sublattices of index n do it. This row is the sublattice count with every entry filled.

p2 — every index. A half-turn preserves every sublattice, because −1 acts trivially on any lattice: the negative of a lattice vector is a lattice vector, always. So p2 is as unconstrained as p1.

pm, pg, pmm, pmg — every index. A mirror or a glide constrains the direction of the thinning but never its amount: scaling along the mirror or across it works for any factor, so every index is reachable even when many sublattices of that index are not.

cm and cmm — every index except 2, 6, 10, 14. The rhombic lattice’s mirrors run along its diagonals, so a sublattice must keep the two equal-length vectors equal; the missing indices are the ones congruent to 2 modulo 4, where the only sublattices of the right shape have the wrong centring.

pgg — the odd indices. A glide’s half-translation must remain half of a lattice vector.

p4, p3, p6 — the norms. The rings supply the answer.

p4m, p4g, p3m1, p31m, p6m — fewer. A mirror on top of a rotation restricts the ideal’s generator to a symmetry direction.

What the round trip checked, and how #

Index three must fail for p4. Three is not a sum of two squares, and if the search found a subgroup there the whole pattern would be wrong.

Index two must fail for p3. Two is not Loeschian.

Index two must succeed for p4, on the sublattice turned by forty-five degrees, which is the positive control: a refusal test that rejected everything would pass the first two and mean nothing.

And index one must be the group itself, for all seventeen — the trivial case, which a search that had gone wrong in its bookkeeping would be as likely to miss as any other.

Why “isomorphic” is the right word and “self-similar” is not #

A group containing a copy of itself sounds like self-similarity, and the two ideas are close enough to be worth separating.

Self-similarity is a statement about a set. A Penrose tiling is self-similar because inflating it by the golden ratio produces the same tiling — the object maps onto itself under a scaling.

An isomorphic subgroup is a statement about a group. The subgroup and the parent are abstractly the same group, and the pattern the subgroup generates is a different pattern from the parent’s — coarser, with fewer points. Nothing maps onto anything.

The two coincide when the scaling that relates the two lattices is a similarity of the plane, which for the square lattice means multiplication by a Gaussian integer and for the hexagonal lattice by an Eisenstein one. That is why the indices are norms: an index is possible exactly when the required scaling is a complex multiplication the lattice permits, and the norm of a complex number is the factor by which it scales areas.

Read that way the whole result is one sentence: a plane group contains a copy of itself at index n exactly when the plane has a similarity of area factor n that preserves the group’s lattice — plus, for the groups with mirrors, that the similarity must preserve the mirrors too.

Where the exactness stops #

A witness, not a count. The search stops at the first sublattice that works, because the question is whether the index is possible. How many isomorphic subgroups a group has at a given index is a different and larger question — it is a count of ideals of a given norm, which for the Gaussian integers is a divisor sum again — and this collection does not compute it here.

The grid stops at sixteen because the enumeration of sublattices grows and the identifier is thorough rather than fast. The patterns are visible well before that and the arithmetic explains them, so extending the grid would confirm rather than inform.

“The same group” means the same plane-group type, which is what the identifier decides. It does not mean the subgroup is conjugate to the parent — it cannot be, since it is smaller — nor that there is a unique way to sit inside it.

The identifier is the weak link and is deliberately the same weak link as everywhere else. Whether the subgroup is “the same group” is decided by a routine that searches a finite list of basis changes and a grid of origin shifts. It is complete for the seventeen because the list and the grid were chosen for them, and a subgroup expressed in a basis outside that list would be reported as unidentifiable rather than as wrong — which is what happened before the sublattice bases were reduced, and is the reason the reduction is described above rather than left as an implementation detail.

And this is the plane. In three dimensions the same phenomenon is tabulated in the International Tables as the isomorphic subgroups of lowest index for every space group, and the arithmetic is correspondingly richer — orders of ideals in more rings, and indices constrained by more than one rotation at once. Nothing here computes that case.

Who names them, and where #

The term isomorphic subgroup is crystallographic rather than mathematical — a group theorist would say a subgroup isomorphic to the whole group and think nothing more of it, since an infinite group containing a copy of itself is unremarkable. What makes it a named object in crystallography is that the copy has the same space-group type, which is a much stronger condition than abstract isomorphism: it requires the same lattice type, the same point group in the same orientation, and the same intrinsic translations.

The International Tables list, for every one of the two hundred and thirty space groups, its maximal isomorphic subgroups of lowest index, and the entries are of exactly this kind: a cell multiplied by a small factor, a symbol unchanged. Billiet and Bertaut worked out the general rules in the 1970s, and the tables’ modern form comes from that work.

Reading a plane-group row of the grid above is reading the two-dimensional version of one of those entries, computed rather than looked up — and the arithmetic reason is the same one that makes the three-dimensional entries look arbitrary until the rings are noticed.

Where the ladder goes next #

An isomorphic subgroup is a superstructure seen from the other end: a crystal whose cell is n times its parent’s has a space group of index n inside the parent’s, and when that subgroup is isomorphic the superstructure has the same symmetry as the structure it orders. Which reflections that adds is a question this collection has already asked, and the answer now has an arithmetic reason attached.

The indices multiply #

There is a structural fact about each row of the grid that the picture makes visible and does not explain: the set of admissible indices is closed under multiplication.

Five and thirteen are both sums of two squares, and so is sixty-five. Three and seven are both Loeschian, and so is twenty-one. That is not a coincidence of small cases — it is the norm being multiplicative: the product of two norms is the norm of the product, which is the Brahmagupta–Fibonacci identity for two squares and its Eisenstein counterpart for the other form.

The consequence for subgroups is immediate and is the reason it matters here. An isomorphic subgroup of an isomorphic subgroup is an isomorphic subgroup, of the product index — so the relations compose, and the admissible indices of a group form a multiplicative semigroup rather than an arbitrary set.

That means each row of the grid is generated by a short list of indices, and the list is not quite the admissible primes. For p4 it is two, the primes congruent to one modulo four, and the squares of the primes congruent to three modulo four — nine is on the row and is a product of neither of the others, because three is not a norm and nine is. For p3 and p6 it is three, the primes congruent to one modulo three, and the squares of the primes congruent to two modulo three. Those generators are the norms of the primes of the Gaussian and Eisenstein integers, and they are exactly the indices at which the copy of the group is maximal. Knowing them gives the whole row, which is why the rows look like number-theoretic lists rather than like lists of indices.

It also says how a superstructure can be reached in stages. A cell enlarged by sixty-five can be reached by enlarging by five and then by thirteen, in either order, through an intermediate structure that is itself a legitimate isomorphic subgroup — and a material with a sequence of ordering transitions can take exactly that route.

The one thing multiplicativity does not give is uniqueness. Sixty-five is a norm in two essentially different ways, 1² + 8² and 4² + 7², so there are two distinct sublattices of index sixty-five keeping the four-fold, and the two are not related by any symmetry of the parent. Composition of relations and multiplication of indices agree; the number of subgroups at an index is a finer count, and it is the number of ways the index factorises in the ring.