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Concept

Automorphism

A symmetry of a structure: a relabelling of its elements that preserves every relation the structure has. The automorphisms of an object form a group under composition, and that group is the precise statement of what the object's symmetry is.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

The subgroups of a polynomial's symmetries, against the fields they name. Two lattices side by side, one of the subgroups of the symmetry group of the cube roots of two and the other of the fields between the rationals and the splitting field, drawn so that one is the other turned upside down.

The lattice that runs the other way

The symmetries of a polynomial's roots form a group, and the fields between the bottom and the top form a lattice. The two are the same picture, one of them turned over — a bigger group of symmetries fixes less, so it names a smaller field.

algebra · Galois correspondence Orbit times stabiliser is 8, on every row. A table of 5 things the 8 symmetries of a 4-gon can move. Each row draws every position the thing can be carried to and lists the motions that leave it where it is; the two counts multiply to 8 on every row.

Twenty-four ways to set a cube down

Count the rotations of a cube from its corners and the answer is eight times three. Count from its edges and it is twelve times two; from its faces, six times four. Three different pictures give one number because each count is the same theorem — the places a thing can go, times the motions that leave it where it is — and the same theorem splits Cayley's sixteen trees into twelve and four and proves that a group of eight has a centre.

algebra · Symmetry groups How often a prime divides the class number. p 3: 39.1% against 1/p 33.3% and predicted 44.0%; p 5: 22.6% against 1/p 20.0% and predicted 24.0%; p 7: 15.3% against 1/p 14.3% and predicted 16.3%; p 11: 9.1% against 1/p 9.1% and predicted 9.9%.

Class groups drawn at random

The class number of an imaginary quadratic field measures how badly unique factorisation fails there, and it looks arbitrary: 1, 2, 4, 3, 5, 12, 1. Count how often 3 divides it and the answer is not one time in three but noticeably more — heading, by a conjecture of Cohen and Lenstra, for 44 per cent. The reason is that class groups behave like finite groups chosen at random with each group weighted by one over its number of symmetries, so groups with few symmetries, like the cyclic group of order 3, turn up more often than the naive count suggests.

number · Unique factorisation The 11 trees on 7 points, and how many labellings each has. All trees on 7 points with their numbers of symmetries and of distinct labellings; the labellings add to 16807.

Almost every tree can be turned over

Cayley's n^(n−2) counts trees with labels on their points. Take the labels off and the count has no formula, because a symmetric shape absorbs labellings: the star on seven points can be labelled only seven ways, the one asymmetric shape 5,040. The bookkeeping that reconciles the two counts says something unexpected — almost every tree, labelled or not, can be turned over onto itself, where almost every graph cannot.

discrete · Labelled trees

Named alongside it

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

Group actionLabelled treeCanonical formClass groupClass numberConjugacy classCosetCounting argumentCounting two waysDegreeDihedral groupExhaustive search

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