Crystallography & Symmetry
patterns derived from their groups
Concept

Abelianisation — where it appears

What is left of a group when every pair of elements is forced to commute, an invariant computed from a presentation by a Smith normal form. It separates p3m1 from p31m, which every picture on this site can only illustrate, and it is the first homology of the quotient when the action is free.

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

p3m1 and p31m, told apart without a picture. The two groups this site returns to most often: same point group, same lattice, same number of operations, and distinguished in every other essay here by where their mirrors sit relative to the lattice — which is a fact about the plane. Abelianised, they are ℤ2 and ℤ6, which are not isomorphic. That difference is a fact about the groups: no change of basis, no redrawing and no relabelling can carry one to the other, and the argument never mentions a mirror line.

What is left when the order is forgotten

Abelianising a group throws away the order of the letters in every word and leaves a small abelian group behind. It is computed by a Smith normal form, it never mentions the plane, and it separates p3m1 from p31m — which a picture can only illustrate.

operations · Presentations Every plane group from at most 4 operations. For each group, the fewest operations that generate the whole of it — the point operations and both lattice translations, since a group that does not reach its own translations is a different group. The floor is the abelianisation's number of invariant factors, which no group can beat, and the search is exhaustive over the operations within one cell of the origin. 14 of the seventeen meet their floor, which settles those exactly; the other 3 need more than the abelian argument can see, and p3m1 needs three where its abelianisation is cyclic.

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.

operations · Presentations Subgroups of index two, three and four. Every plane group with the number of subgroups it has at each small index, counted by enumerating the transitive actions on that many points. The zeros are the interesting entries: p3 has no subgroup of index two and the four-fold groups have none of index three, because a subgroup of index n gives an action on n points and the group has to have a quotient that can act. A rotation of order three has nowhere to go in a set of two, and one of order four has nowhere to go in a set of three that is not the identity — so the index is constrained by the point group before any geometry is done.

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.

operations · Subgroups The ball of radius 5 in p6. Every element of p6 reachable in at most 5 multiplications by a generator or its inverse, plotted at its translation part — so each dot is a lattice position and its size says how few steps reach it. The picture is the word metric's unit ball scaled up, and its shape is what fixes the growth: a diamond where the group supplies two short translations, and a hexagon where it supplies three. Every dot here required the word problem to be solved, because the search has to know when two products are the same element.

Telling two words apart

There are finitely presented groups in which no algorithm can decide whether two products of the generators are the same element. The seventeen are not among them, and the procedure that settles it is short enough to state in a sentence — which then makes it possible to measure how fast each group grows.

operations · Presentations p1 folds into a torus. The cell of p1 with its edges marked as the group joins them: both pairs by a plain translation, both arrows the same way round. Gluing top to bottom gives a tube and gluing its ends gives a torus. Nothing in p1 holds a point still, so the surface has no marked points and its first homology is two copies of the integers.

The two that fold into a surface

Fold a wallpaper pattern along its own symmetries and what is left is usually a shape with corners and edges nobody drew. For two of the seventeen it is a plain surface with no marks on it at all — a torus and a Klein bottle — and which two is decided by a single question asked of every operation.

classification · Flat space Every subgroup of index two is normal; at index three most are not. For each of the seventeen plane groups, its abelianisation and the number of normal subgroups of each small index against the number of subgroups of that index. The index-two column is complete every time, because the left and right cosets of a subgroup of index two are the same pair of sets. At index three and four the two numbers part, and the gap is what normality costs: a subgroup that is carried to a different subgroup by some operation of the group it sits in.

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.

operations · Subgroups Four of the seventeen have a centre, and they are the four with no rotation. For each plane group: the order of its point group, how many of its operations are rotations, the lattice vectors every operation of the point group fixes, and the centre those vectors make. A central element must commute with every translation, which forces its linear part to be the identity — so the centre is a group of translations, and a translation is central exactly when the point group leaves it alone. A rotation leaves nothing alone but zero.

The four groups with a centre

An element that commutes with everything has to commute with every translation, and that forces its linear part to be the identity. So the centre of a plane group is a group of translations — the ones its point group leaves alone — and a rotation leaves nothing alone but zero. Four of the seventeen have a centre and thirteen have nothing at all.

operations · Composition

Named alongside it

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

PresentationCosetInvariant factorNormal subgroupRelatorSmith normal formSubgroupTranslation groupConjugationFixed pointIndexOrbifold

All concepts