Subgroup — where it appears
Named by 38 essays across 7 fields — each of them below, with the objects they name alongside it.
Thirty-two, and no others
There are exactly thirty-two ways a crystal can be symmetric about a point. Not thirty-two that anybody has catalogued — thirty-two that a finite search produces, from two starting groups, with every step of the reduction counted separately so that no two of them can quietly compensate.
3m1 and 31m are one class
This site has an essay arguing that p3m1 and p31m are genuinely different groups. As point groups the same two objects are one class — and the two subgroups are each normal in the hexagonal holohedry, so nothing in the lattice relates them. What does is a rotation of thirty degrees.
The holohedry is the ceiling
A crystal never has more point symmetry than its lattice. That single containment decides which system a class belongs to, why there are seven systems and not thirty-two, and why a lattice can be more symmetric than the crystal sitting on it — which is the usual case rather than the exception.
Domains of a subgroup
A group with half the operations needs twice as much of the cell to rebuild the pattern from. That single sentence is the index arithmetic of the whole classification, and it turns the containments among the seventeen into a statement about area.
The descent of symmetry is a lattice, not a tree
Which classes a crystal can fall to when it loses symmetry, drawn as a graph with the index on every edge. It is routinely called a tree and it is not one — a class can be reached from its parent by several different routes of the same total index, and which route a material takes is a physical question the diagram deliberately leaves open.
The friezes inside the seventeen
Take one lattice row of a wallpaper pattern and keep only the symmetries that leave that row where it is. What survives is a frieze group — and which of the seven it turns out to be is a fact about the plane group that its symbol does not state.
Forty-eight becomes sixteen
Centre one face of a cube and the four threefold axes along its body diagonals are gone. That sentence is usually offered as a fact to accept; it is a computation whose answer is a number, and the number says which lattice you got instead.
Two ways down from a group
A pattern can lose a symmetry by giving up an operation or by giving up a translation, and the two are different in kind. Sorting the seventy-four subgroups of index two among the seventeen splits them twenty-nine to forty-five — and a containment test that compares operations modulo one shared lattice can only see the twenty-nine.
Two colours, and a symmetry that swaps them
A chessboard and a grid of identical squares have the same group, which is plainly not what anybody sees. Admitting the colour swap as an operation gives a finer classification — and one of the seventeen turns out to admit no two-colouring at all.
Two hundred and forty-seven descents, or two hundred and twelve
How many distinct ways can a crystal lose symmetry? Counting parent-and-child pairs up to conjugacy in the parent gives 247. The standard enumeration in the ferroics literature gives 212, and the operation that merges the extra thirty-five turns out to be a rotation through forty-five degrees — which no lattice may have, and which no integer matrix in a lattice basis can therefore express.
The most of an icosahedron a crystal can keep
C₆₀ sits in crystals and virus capsids sit in crystals, and neither of them stops being icosahedral. What a lattice can fix is a subgroup — and the largest crystallographic subgroup of the sixty rotations has order twelve, at index five. The five are Kepler's five cubes.
How few operations make a pattern
A plane group is infinite, and a handful of its operations is enough to rebuild all of it. How small a handful is a question with a floor from the abelianisation and a ceiling from an exhaustive search, and for fourteen of the seventeen the two numbers meet.
How many subgroups of index three
Taking operations away and closing what is left finds the maximal subgroups and stops there. Counting instead the ways a group can act on three points finds all of them — and finds that a four-fold group has none of index three at all.
What a molecule gives up to sit in a crystal
A molecule brings its own symmetry. A crystal offers sites with symmetries of their own, and the two have to be compatible — the site's symmetry must be a subgroup of the molecule's. So a molecule may always keep more than its site offers, and a molecule with a five-fold axis may sit only where the crystal offers nothing at all.
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.
Ten ways for space to be flat
Thirteen of the two hundred and thirty space groups hold no point still, and folding space along one of them gives a shape with no curvature anywhere. There are ten such shapes, not thirteen, and the difference is the same eleven pairs that separate 230 from 219.
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.
What a crystal keeps in a field
Curie's principle says the symmetry of an effect contains the intersection of the symmetries of its causes. Applied to a crystal in a field that is an intersection of two groups, one of them infinite — and it comes out exactly, class by class, as a subgroup that decides which effects are permitted next.
The quotient each normal subgroup leaves
Two hundred and eighty-one subgroups of index four across the seventeen plane groups, and ninety-seven of them normal. Which ones, and what is left when they are divided out, needs no enumeration at all: below order six every group is abelian, so a normal subgroup of small index is a subgroup of the abelianisation and its quotient is decided by a product of greatest common divisors.
What a half-turn does to three colours
Ten of the seventeen plane groups have no three-colouring, because a half-turn cannot permute three colours cyclically, and that is true. Drop the word cyclically and the answer changes completely: a half-turn permutes three colours perfectly well by swapping two and fixing one, and only the three four-fold groups refuse three colours at all.
Thirty-two from fourteen matrices
Write the fourteen Bravais lattices as Gram matrices, ask each one which integer matrices preserve it, and take every subgroup of every answer: five hundred and ten of them. Sort those by how many operations of each kind they contain — which a determinant and a trace decide — and thirty-two answers come out. They are the crystal classes, from a construction in which no point group is ever named.
Seventy-three, without a search
The unit a space group is built from is a point group together with the lattice it acts on, and there are seventy-three of them. Getting there looks like it needs conjugacy in GL(3,ℤ), which is a search this collection tried and abandoned. It does not: every finite group of integer matrices carries a canonical larger group that says which lattice it belongs to, and once that is computed the search has nothing left to do.
How many axes there are is a Sylow count
Sylow's theorems say that the subgroups of prime-power order in a finite group are all conjugate and that how many there are is congruent to one modulo the prime. Applied to the thirty-two crystal classes that arithmetic counts axes: the number of Sylow 3-subgroups is the number of equivalent threefold directions, it is four in exactly five classes, and those five are the cubic ones. So cubic has a definition with no geometry in it.
The table that decides every action
Burnside's lemma counts orbits and stops there — two completely different actions with the same orbit count are indistinguishable to it. The object that settles the whole question is a square table whose entries count fixed cosets: lower triangular because a subgroup fixes no coset of anything smaller, positive on the diagonal because it fixes its own, and therefore invertible. Inverting it turns a list of fixed-point counts back into the orbits themselves.
How many orientations a disorder needs
A molecule at a site with more symmetry than it has resolves the contradiction by occupying several orientations at once. How many is not fitted: it is the index of the molecule's symmetry in the site's, so the occupancy is the reciprocal of a whole number — and one site in the census refuses a divisor of its own order.
The occupancy does not name the disorder
A molecule disordered on a special position takes a number of orientations fixed by a group index, and its occupancy is the reciprocal. Many different disorders share one occupancy — eleven at a single kind of tetragonal site — and what separates them is which of the site's operations the molecule keeps, which the averaged structure records and the occupancy does not.
Three of them, and they are equivalent
The subgroup tables print a count and sometimes a word beside it. Three subgroups of one type may be three copies the group itself shuffles, or three the group holds firmly apart and only a change of description exchanges. p3 has three copies of itself at index three, no operation of p3 moves any of them, and one shift by a third of a cell exchanges all three.
Going up costs the cell a parameter
The usual asymmetry — finitely many maximal subgroups below, infinitely many minimal supergroups above — is false in both halves for a plane group. Both directions are infinite and equinumerous index by index. The real asymmetry is that 17 of the 31 edges cost the lattice a parameter going up and nothing going down.
Two conditions are not one condition twice
Whether a plane structure has a higher-symmetry supergroup is decided by its measured cell, and the obvious measure — how many standard uncertainties the cell sits from a right angle, or from equal edges — cannot rank the candidates on its own. A square needs two conditions and a rectangle one, so a single distance means different things on each; counted properly, a square can score above the equal-edged lattices that contain it, and walking up the diagram one condition at a time is the reading that keeps the ranking coherent, at a price that has to be paid in its error rate.
Four groups are made of mirrors
Close the plane from located mirrors instead of rotation centres and the closure is discrete only for four triangles of mirrors — and one of those four manufactures a mirror along its own altitude. Of the ten plane groups that contain a reflection, exactly four are generated by their mirrors; three hold a kaleidoscope at index two, three and four, which are the numbers in front of the star in their orbifold symbols; and three have mirrors in one direction only, which generate nothing along themselves.
Six groups are made of glides
Four plane groups are generated by their mirrors. Six are generated by their glides, and they are mostly different ones, because a glide done twice is a translation where a mirror done twice is nothing. Closing the plane from located glide lines with their slides reaches six of the seventeen; mirrors and glides together reach eleven, every group with a reflection except pm. The classification does not fall out of the closure, and where it stops says what each kind of reflection can and cannot make.
Cutting a sheet can give it a hand
Cut a periodic sheet along a line and the edge that is left has a symmetry of its own: a frieze group, drawn in the plane across the cut. A chiral sheet can only have chiral edges. An achiral one usually has chiral edges too, in all but a few directions — and eight of the eighty layer groups have no achiral edge at all, however the sheet is cut.
Ninety-six four-colour designs
The seventeen plane groups have 281 four-colourings, counted as subgroups of index four, and many of them are the same design described from a different origin, on different axes or from a neighbouring cell. Identified under every affine map that carries the pattern's group to itself, they make ninety-six designs. The same computation run at two and three colours gives forty-six and twenty-three, and all three numbers match the classical counts of coloured plane groups.
Seven space groups cannot make their own translations
Every space group contains a lattice of translations, and for all but seven of the 219 types those translations are products of the group's other operations. A screw done twice climbs its axis and a glide done twice slides along its plane, so the non-symmorphic operations make lattices by themselves. The seven that cannot are exactly the seven plane groups with no glide, repeated along a line they all fix, and the reason is a difference between a glide line and a glide plane.
Two forty-sixes that do not match
The layer groups contain forty-six two-coloured plane patterns, and the rod groups contain forty-six threads that turn their axis over. Both lists are lists of a group with a half of index two, and both halves have a point group in the plane, so they can be laid side by side. They do not line up on any invariant tried: the plane spends its forty-six on rectangular mirrors, which halve a lattice easily, and the rods spend theirs on high rotations, which carry the most kinds of screw. The equal totals are two sums that happen to meet.
A cubic cell has fourteen chances to fail
In the plane, deciding whether a measured cell is square meant two steps up a short diagram, and dividing the error rate between them fixed the test. In space, a primitive cubic cell sits on ten strata joined by fourteen steps on ten different paths, and a walk that tests every step at five per cent calls more than a quarter of true cubic cells something less. Dividing the level by the steps of one path is not enough and dividing it by all fourteen is too much. Measured directly, the right level is two-thirds of a per cent a step, and calibrated that way the walk notices a tetragonal distortion more often than the single test it replaces.
The centring survives the limit
Stretch a body-centred tetragonal lattice until its axes are equal and it becomes body-centred cubic; stretch it further and it becomes face-centred cubic. No stretching of a primitive tetragonal lattice reaches either. Which of the fourteen lattices can be deformed into which is decided by the subgroups of each lattice's own group, and the answer keeps track of centring in a way the familiar chain of crystal systems does not.
No cell joins the seven halves
Every identification the normaliser makes between Wyckoff positions is made by an isometry of some cell, and it was natural to expect the same of subgroups. It nearly is. Across all 219 types and their 1,813 subgroups of index two, one cell does everything the normaliser does, except in the two triclinic groups. P1 has seven ways to be halved, and its normaliser makes them all one. No cell can, because seven is the number the crystallographic restriction forbids.
Named alongside it
The objects these essays reach for when they reach for this one.
IndexHolohedryConjugacy classCosetCrystal classPoint groupNormaliserCrystallographic restrictionMetric tensorOrbitPlane groupSite symmetry