Matpic
mathematics, in pictures
Concept

Generating set

A collection of elements whose products reach every element of a group. The choice is not part of the group, so a picture drawn from it carries features that are artefacts — the degree and the diameter — beside features that are not, such as the growth rate.

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

The dihedral group of a 4-sided shape, drawn as a map. A Cayley graph: one dot per motion of the shape, with one arrow per generator, so that multiplying by a generator is following an arrow of that colour.

The group drawn as a map

A multiplication table says everything about a group and shows nothing; lay the same information out as one dot per element and one arrow per generator, and multiplying becomes walking, distance becomes a word length, and the group acquires a shape.

algebra · Cayley graph The ball around the identity, in 3 groups. A table of the number of group elements within each distance of the identity, one row per group, with the growth type each row exhibits beside it.

How fast the ball fills

Count the elements within r steps of doing nothing. The count grows like a polynomial in some groups and like a power of three in others, the distinction survives every change of generating set, and which polynomial degrees are possible is a theorem nobody expected.

algebra · Cayley graph

Named alongside it

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

Cayley graphCyclic groupGroup actionWord metricCosetDihedral groupFree groupGrowth rateInvariantSymmetry

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