Crystallography & Symmetry
patterns derived from their groups
Concept

Schreier lemma — where it appears

The rule giving generators for a subgroup from a transversal, turning a search for what a set generates into a finite list. Without it, finding a subgroup's own presentation from its parent's would be a search with no stopping condition.

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

p4g in 4 letters and 8 relations. The presentation of p4g, derived from the group's own operations. The two translations commute; each conjugation relation is read off a column of a matrix; and the point group's relations are corrected by the translation they actually come back as, which is what makes this group an extension rather than a semidirect product. Every relator is evaluated where the group lives and must be the identity, and coset enumeration on the letters alone returns 8, which is the order of the point group.

A group in four letters

Every other essay here describes a symmetry group by what it does to the plane. There is a second description — a handful of letters and the words in them that are required to equal nothing — and it can be counted with no plane anywhere in the computation.

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

Named alongside it

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

CosetGeneratorsPresentationAbelianisationClosureCoset enumerationDecidabilityGroup extensionHalf-turnHermite normal formInvariant factorRelator

All concepts