Crystallography & Symmetry
patterns derived from their groups
Concept

Klassengleiche — where it appears

Said of a subgroup keeping every linear part of its parent and giving up translations, so that the cell grows instead. A containment test that compares operations alone cannot see this kind, which is why counting subgroups needs more than a closure.

Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.

The subgroups of p4m of index 2. p4m has 7 subgroup(s) of index 2 with cyclic quotient. 3 of them keep every translation and lose operations — the lattice is untouched and the pattern loses a symmetry at every point. 4 keep every operation and lose translations, and each is named beside the basis of the sublattice it keeps, written in the parent's own axes. Each subgroup is the kernel of a homomorphism onto a cyclic group, found by enumeration; each name is found by searching changes of basis and origin until the operation sets match exactly.

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.

operations · Subgroups The seventeen, arranged by what they can lose. Each group at the height of its own order, joined to every maximal subgroup that keeps all of its translations. Reading downwards is a crystal losing operations at a phase transition. The edges are the maximal ones only — every other containment is a path through these — and the whole graph is enumerated by closing every subset of each group's operations, so nothing is here because a table said so.

The descent with no shortcut

A subgroup can give up operations, or it can give up translations. Hermann's theorem says that a *maximal* subgroup does one or the other and never both at once — which is why a crystal losing symmetry can be followed one clean step at a time, and why every route from p6m down to p1 has exactly three steps.

operations · Subgroups 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.

space-groups · Isomorphic subgroups p4m: freezing Γ3 leaves pmg. The same crystal three times. On the left, a pattern with the full symmetry of p4m. In the middle, the displacement each atom takes under the order parameter — the arrows are the mode, drawn in the first colour for one set of atoms and the second for the other where there are two. On the right, the atoms moved by a small multiple of those displacements. The group of the right-hand pattern is pmg, of index 4 in the parent, and it was found by the detector from the point set alone. The prediction — which operations carry the displacement field to itself — is made separately, from the mode and not from the points, and the two lists of operations are identical.

The cell a zone-boundary mode doubles

An order parameter that alternates from cell to cell keeps only half the translations, so the frozen structure has a cell twice as large and reflections that were never there before. The phases at such a wavevector are ±1, so the whole computation stays in exact integers.

space-groups · Landau 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.

space-groups · Isomorphic subgroups Subgroups, the classes a group sorts them into, and the sets its normaliser does. For every plane group, the number of subgroups of index two and of index three, the number of conjugacy classes those fall into under the group's own operations, and the number of sets they fall into under its Euclidean normaliser. Over the seventeen there are 74 subgroups of index two in 74 classes and 56 sets, and 82 of index three in 36 classes and 32 sets. 9 of the thirty-four rows have fewer sets than classes, which is where the tables' "equivalent" entries come from. Counts of subgroups and of classes agree with an independent count from transitive actions on n points.

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.

operations · Normalisers Free going down, two conditions going up. The edge between p2 and p4 in the diagram of maximal translationengleiche relations, read in both directions. Downwards it costs nothing: the quarter-turns are discarded and the lattice is exactly the lattice that was there, so every p4 pattern contains a p2 pattern. Upwards the added quarter-turns must carry the lattice onto itself, which forces the cell to have equal edges at a right angle — two conditions on a general oblique cell, which has only two parameters to give. So a p2 structure has a p4 supergroup exactly when its measured cell happens to be square, and the question is about the metric rather than about the group.

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.

operations · Subgroups

Named alongside it

The objects these essays reach for when they reach for this one.

IndexSubgroupSublatticeMaximal subgroupTranslationengleicheGaussian integerIsomorphic subgroupLoeschian numberPlane groupSuperstructureAntiphase boundaryComposition series

All concepts