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Concept

Alternating group

The group of permutations of a finite set that can be built from an even number of swaps, exactly half of all of them. It is the rotation group of the tetrahedron at four letters and of the icosahedron at five, and on five or more letters it cannot be broken into smaller pieces.

Named by 7 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 36 3-tuples with product e, and the 3 that no turn moves. Every ordered choice of 3 elements of the 6 symmetries of a 3-gon whose product is the identity, in cards grouped by cyclic turning. 3 cards hold a single tuple repeating one element; the other 11 hold 3 each.

Necklaces made of symmetries

Lagrange's theorem says a subgroup's size divides the group's, and the converse is false. One piece of the converse is true: every prime that divides the size is the order of some element. The proof threads the group's own elements onto a necklace whose product is nothing, turns it, and counts — the argument that proved Fermat's little theorem with beads, with the beads replaced by motions.

algebra · Symmetry groups The twenty maps x ↦ ax + b modulo 5, as permutations of five roots. A four-by-five grid of pentagons; in each, arrows show where the map ax + b modulo 5 sends each of the five corners. Rows are the slopes 1, 4, 2, 3; columns the shifts 0 to 4.

The quintics that have a formula

No formula solves every equation of degree five, and yet x⁵ − 2 is solved by a fifth root and x⁵ − 5x + 12 by a longer expression of the same kind. A quintic can be solved by radicals exactly when the symmetries of its roots fit inside one group of twenty — the maps x ↦ ax + b on the numbers modulo 5 — and two exact tests on its coefficients say whether they do.

algebra · Galois correspondence Every pair of rearrangements of four letters, and what the two generate. A 24 by 24 grid of ordered pairs of permutations of four letters: 216 generate S4, 96 generate A4, 264 generate a smaller subgroup.

Two random shuffles reach every shuffle

Pick two ways of rearranging n letters at random and allow them to be repeated in any order. Almost always the two together reach every rearrangement there is, or every even one — and when they fail, the usual reason is a coincidence about a single letter that both happen to leave alone. Eugen Netto guessed it in 1882 and John Dixon proved it in 1969.

algebra · Symmetry groups The 3,600 pairs of icosahedral rotations, and what each pair generates. A 60 by 60 grid of ordered pairs of elements of A5 shaded by the order of the subgroup they generate; 2280 generate A5.

Nineteen copies and no more

Two of the icosahedron's rotations, chosen at random, generate all sixty of them nineteen times in thirty. Philip Hall found the number in 1936 by inclusion and exclusion over every subgroup at once, weighted by the Möbius function of the subgroups' order — and the same count says that nineteen copies of the group, side by side, can still be generated by two elements, while twenty cannot.

algebra · Symmetry groups Even quartics never have the full group, and one step away almost all do. Even quartics x^4 + bx^2 + d to 20: reducible 199, S4 0, A4 0, D4 1338, C4 20, V4 124. With +x: reducible 157, S4 1524, A4 0, D4 0, C4 0, V4 0.

The quartics whose roots stay paired

A quartic can have five different Galois groups, and in a box of nearly fourteen million quartics they do not come in order of size. The eight-element group D₄ outnumbers the twelve-element A₄ thirty-five to one. Every even quartic has a smaller group than S₄, and moving one step off that plane restores the full group almost everywhere.

algebra · Galois correspondence

Named alongside it

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

PermutationSimple groupConjugacy classDihedral groupDiscriminantGalois groupGenerating setProbabilityRadical extensionSolvable groupSubgroupCommutator

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