Concept

Commutator

The element aba⁻¹b⁻¹ of a group, which is the identity exactly when a and b commute. It measures how far two elements are from commuting, and the subgroup all the commutators generate is the smallest one whose quotient is abelian.

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

the Borromean rings, with every crossing signed. A diagram of the Borromean rings with the under-strand broken at each crossing and each crossing between two components marked with its sign, which add to twice the linking number.

Linked, and no two of them are

Three rings that cannot be pulled apart, in which every pair comes apart the moment the third is removed. Every pairwise linking number is zero, so the number cannot see it — and what does see it is a word in two letters that refuses to cancel.

topology · linking number

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

Named alongside it

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

Alternating group Borromean rings Brunnian link Conjugacy class Derived series Free group Fundamental group Linking number Milnor invariant Normal subgroup Radical extension Simple group

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