Concept

Simple group

A group with no way of being broken into a smaller group and a quotient, other than trivially. The smallest non-abelian one is the rotation group of the icosahedron, and its indivisibility is why the general fifth-degree equation has no formula in radicals.

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

The derived series of S3, S4, S5. A table with one row per group giving the sizes along its derived series, each step the subgroup generated by all commutators of the last, and whether the series reaches the identity.

The group that will not come apart

Solving an equation by radicals means building a tower of roots, and a tower of roots corresponds to a chain of subgroups with abelian steps. For the general equation of degree five that chain would have to descend through a group of sixty elements with no normal subgroup in it — so there is no formula, and the obstruction is a finite object that can be written out.

algebra · Galois correspondence

Every turn that leaves a cube where it was. A cube in wireframe beside a table of its rotation axes: 3 of order 4, 4 of order 3, 6 of order 2, totalling 24 turns including the one that does nothing.

The five solids as three groups

There are five regular solids and only three groups of rotations between them, because a solid and its dual share their symmetries exactly. The largest of the three is the smallest group with no way of coming apart, which is why the general equation of the fifth degree has no formula.

geometry · Regular polyhedra

Named alongside it

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

Alternating group Conjugacy class Commutator Derived series Duality Group order Normal subgroup Permutation Platonic solids Radical extension Rotation Solvability

All concepts