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 2 essays across one field — 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

Named alongside it

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

Conjugacy class Coset Counting argument Degree Dihedral group Field extension Fixed field Galois group Group action Index Labelled tree Lagrange theorem

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