Isomorphic subgroups — the series · Crystallography & Symmetry

At which indices a group contains a copy of itself. A filled circle where the group has a subgroup of that index which is the same plane group again. The groups with no rotation past a half-turn take every index — the lattice can be stretched along one direction by any factor. The four-fold groups take the sums of two squares and the three- and six-fold groups take the Loeschian numbers, because a sublattice invariant under a quarter or a third of a turn is an ideal in the Gaussian or Eisenstein integers and its index is a norm. The groups with mirrors take fewer still, and p4g takes only the squares.

The same group in a bigger cell

A subgroup usually gives something up. An isomorphic subgroup gives up nothing but scale — the same plane group again, on a coarser lattice — and the indices at which that is possible turn out to be the values of a quadratic form.

part 1 · space-groups

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.

A bigger cell, and sometimes the mirror

An isomorphic subgroup gives up nothing but scale — the same group again on a coarser lattice. In space the screw axes sharpen the question, and the answer contains a surprise: a cell three times taller holds the group's enantiomorphic partner, so a left-handed screw contains a right-handed one with nothing done to the crystal but a change of description.

part 2 · space-groups

Which indices a screw axis contains itself at. The fifteen kinds of axis a space group may have, against the index of the sublattice taken along the axis. A filled cell is an index at which the axis contains a copy of its own kind; the darker cells are the indices at which what comes back is the mirror image instead. A pure rotation axis is filled everywhere and a screw is not, and which indices a screw loses is decided by one congruence rather than by any geometry.

A screw that contains its own mirror image

No proper motion turns a right-handed screw axis into a left-handed one; only a reflection can — that is what makes the eleven enantiomorphic pairs pairs. And yet a 4₁ axis contains copies of 4₃ as subgroups, at every index congruent to three modulo four. One congruence decides both which indices are possible and which hand comes back, and it is the same congruence for all fifteen kinds of axis.

part 3 · space-groups

What lies between a group and its copies at index 25. For each group, the lattices carried to themselves by its point group that contain a copy of the group at index 25, arranged by index, with lines for containment; each lattice is labelled by the Gaussian or Eisenstein integer that generates it. Copies are the bottom row, drawn large when nothing lies between them and the whole lattice. p4: 3 copies at index 25, 0 maximal, with invariant lattices between of index 5; p4m: 1 copies at index 25, 1 maximal, with invariant lattices between of index none.

The primes a cell can grow by

A plane group contains copies of itself in bigger cells, and the International Tables list the ones that are maximal — the copies nothing else sits between. For p4 they come at 2, at 5, 13, 17 and 29 twice each, and at 9 and 49 once, and the list is the list of primes of the Gaussian integers. Put mirrors on the pattern and the copies at 5 and 13 vanish while 25 becomes maximal: a mirror cannot keep one factor of a prime without the other.

part 4 · space-groups

P4₁: the lattices, as an ideal across and a multiple along. Every sublattice the point group of P4₁ carries to itself, indexed by the norm of the ideal it uses across the axis and by the multiple it takes along it. The entry is the space group that sits on it: the parent's own type in one colour, a different type in the other, and a dash where no group with the parent's point group survives at all. A dot marks a lattice that is maximal — one whose step is a single prime, across or along, with nothing between it and the whole. The rows and columns are two divisibility orders and the table is their product, which is the shape the plane's answer predicted.

An ideal across and a prime along

In the plane a copy of a group inside itself grows by a prime ideal, and the maximal indices are the norms of the primes of a ring. In space with one principal axis there are two directions to grow in, and the question the plane left was whether the two constraints multiply. They do not — and the place they fail is an index the plane calls maximal, because the step in between carries the group's mirror image.

part 5 · space-groups

A row of coefficients for every group. For eight of the seventeen, the number of sublattices of each index that the group's point group carries to itself. A nought means the group has no copy of itself at that index at all. p1 and p2 preserve every sublattice, so their rows are the counts of sublattices themselves — 1, 3, 4, 7, 6, 12 — which is the sum of the divisors. The rows thin out as the point group grows, and p4m and p6m have almost nothing in them. Every row is a sequence a Dirichlet series can be built on, and the next figure is what that series factors into.

A row written as a product

Every group's copies of itself sit at a row of indices, and every row so far has been read one entry at a time. Counting all of them at once turns a row into a Dirichlet series, and every one of the seventeen rows factors into a product over the primes — which is the statement that a copy is a chain of maximal steps, written as arithmetic. The plainest group of all has the most famous series in mathematics.

part 6 · space-groups

One lattice, two copies of pm. pm on a lattice doubled across its mirrors. The mirrors of the parent are every vertical line; a copy of pm on the doubled lattice has mirrors every other line, and there are two ways to choose which — the solid set or the dashed set. Both are copies of pm with the same lattice and the same point group, and no translation of the parent carries one onto the other, because the translation that would is exactly the one the doubling removed. So a count of invariant lattices is not a count of subgroups, and the gap is visible in the smallest case there is.

A lattice is not a subgroup

Every count of copies so far has counted lattices, and the International Tables count subgroups. One invariant lattice can carry several copies of a group that nothing in the parent carries onto one another — pm's doubled lattice carries two, with its mirrors on the even lines or the odd ones — and how many is a cohomology computation, a first where the classification of the seventeen used a second.

part 7 · space-groups

Seven indices in 40, and one lattice at each. For every index to 40, how many sublattices of a cubic lattice there are and how many of them the full cubic point group carries to itself. The first number runs into the hundreds; the second is nought at almost every index and one at 1, 2, 4, 8, 16, 27, 32. A cubic point group is forty-eight conditions on a sublattice, and forty-eight conditions leave very little. The richest point group in three dimensions has the poorest arithmetic of copies, and the two are the same fact.

The richest group has the poorest arithmetic

A plane group with no symmetry has seven hundred and sixty-two sublattices to grow by; one with a six-fold axis and mirrors has eight. Take the trend to its end in space and a cubic group has six to index forty — one at every cube, one at twice a cube, one at four times a cube, and nothing anywhere else. Every operation of a point group is a condition, and forty-eight conditions leave almost nothing.

part 8 · space-groups

Copies of a plane group of itself, counted to index ninety-six. For eight of the seventeen plane groups, the number of subgroups isomorphic to the group, up to conjugacy, of index at most N, on logarithmic axes. The curves fall into four tiers by their slopes: groups with no operation but a half-turn grow as N², groups with mirrors and nothing of order three or more as N log N, groups with a rotation of order three or more and no mirror as N, and groups with both as √N. By index 96 p2 has 13446 copies and p6m 13.

The copies a group keeps of itself grow at four speeds

A plane group's copies of itself sit on the sublattices all its operations keep, several to a lattice. A richer point group keeps fewer lattices and puts more copies on each, and the two trends run in opposite directions. Counted to index ninety-six they do not balance. The copies per lattice never exceed sixteen, while the lattices grow as N², N log N, N or √N according to the point group, so a richer group's early lead always ends, at index ten, eleven or twenty-seven, and never returns.

part 9 · space-groups

Sublattices a space group's point group keeps, counted to index sixty-four. For six of the seventy-three arithmetic classes, the number of sublattices of ℤ³ of index at most N that every operation of the point group carries to itself, on logarithmic axes. The six curves have six slopes: N³ for a triclinic class, N² for a monoclinic one, N log² N for an orthorhombic one, N log N for a tetragonal class with no two-fold axis or mirror through its four-fold axis, N for one with them, and the cube root of N for a cubic class. By index 64 P1̅ keeps 176874 sublattices and Pm3̅m 9.

In space the sublattices a group keeps grow at six speeds

The plane's seventeen groups kept their sublattices at four rates. Counted to index sixty-four for all seventy-three arithmetic classes, space has six: the cube of the index, its square, the index times two logarithms, the index times one, the index, and its cube root. Four systems keep one rate each. The tetragonal, hexagonal and rhombohedral classes are cut in two, by whether anything turns their axis over. And the richer groups' lead lasts much longer than in the plane, because they carry up to five hundred and twelve subgroups on a single lattice.

part 10 · space-groups

Where the two crossings the census could not reach fall. The ratio of cumulative subgroup counts with the full point group, poorer class over richer, against the largest index. P1̅ passes Pmmm at 20, as the census to sixty-four found. P2/m passes Pmmm at index 81 and stays ahead to 1,023. P4/m passes P4/mmm at index 12,104 and stays ahead to 65,536. Every count comes from the values at prime powers, multiplied.

The last two crossings fall at eighty-one and twelve thousand

A census to index sixty-four found that P2/m and P4/m must eventually overtake Pmmm and P4/mmm in the number of subgroups they keep of themselves, and could not see either happen. The count turns out to be multiplicative: a sublattice splits into pieces at the primes dividing its index, and so does the cohomology on it, which lives only at two. Every odd prime is then a closed form, two is a short computation, and the crossings fall at index 81 and at index 12,104.

part 11 · space-groups