Crystallography & Symmetry
patterns derived from their groups
Concept

Smith normal form — where it appears

The diagonal form an integer matrix reduces to by integer row and column operations, whose entries successively divide one another. It is what turns a presentation's relation matrix into the list of cyclic factors an abelianised group has.

Named by 4 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 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 Seventy-three classes, seventy-three Smith forms. Every arithmetic crystal class of space, grouped by lattice system, with the cohomology group read off the Smith normal form of the integer conditions its Cayley graph imposes, shaded by the group's exponent. Twelve classes have nothing; fifty-three have exponent two, among them the orthorhombic P holohedry with ℤ2 to the sixth; three have exponent three, three have exponent four, and two — the hexagonal 6 and 622 on the primitive lattice — have exponent six. Nothing larger occurs.

Seventy-three Smith forms

The cohomology group that counts a crystal class's intrinsic translations has a closed form only when the point group is cyclic. For every other class it is still one integer matrix away: the conditions the Cayley graph's cycles impose, reduced to Smith normal form, leave the group on the diagonal. Run on all seventy-three classes of space, the diagonal says the largest exponent is six, that a group of order forty-eight needs nothing finer than halves, and that no class needs a denominator its operations do not already have.

classification · Cohomology

Named alongside it

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

AbelianisationInvariant factorCosetGroup extensionPoint groupPresentationSubgroupTranslation groupArithmetic crystal classCayley graphClosureCoboundary

All concepts