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Concept

Conjugacy class

The set of elements of a group that are the same operation seen from different positions, obtained from one of them by composing with every element and its inverse. Classes partition a group, and a subgroup that is a union of them is exactly one that can be quotiented by.

Named by 3 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 The half of 24 permutations that commutators reach. A block of 24 squares, one per permutation, with the 12 generated by commutators shaded, beside bars counting the homomorphisms to each cyclic group.

The only bit that survives

A shuffle can be called even or odd, and the label behaves under composition. Ask whether some cleverer label — a number out of three, or out of four — could behave the same way, and the answer is that nothing else can — one bit is exactly what a permutation gives up.

algebra · Permutation parity

Named alongside it

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

Alternating groupCommutatorNormal subgroupPermutationSimple groupAbelian groupDerived seriesDualityExhaustive searchGroup orderHomomorphismInvariant

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