Conjugacy class
Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.
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.
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.
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.
When the label may be a matrix
A permutation carries exactly one bit into any commutative target, and commutativity is the restriction doing all the work. Drop it — let the label be a matrix — and what survives is a short finite table, computed here from traces and checked for orthogonality over every pair of rows.
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.
Five-eighths of the pairs, and no more
Pick two symmetries of a square at random and do them in both orders: forty times in sixty-four the result is the same. No group that fails to commute does better. The reason is a count of pairs that turns into a count of conjugacy classes, and a two-line argument about the centre that caps the answer at five-eighths — reached by the square and the quaternions, approached from above by nothing, and approached from below by groups that commute a little more than half the time.
The lists no covering can have
A covering of the sphere by itself with three branch points is three permutations of the sheets whose product is the identity. The Riemann–Hurwitz count says which lists of cycle shapes might work, and a search through every permutation says which do. Up to thirteen sheets, 150 lists pass the count and fail the search — and not one of them has a prime number of sheets.
Counting the subgroups of a free group
The free group on two letters has one subgroup of index 1, three of index 2, thirteen of index 3, then 71, 461, 3,447 and nearly thirty-two million of index 10. Each is a covering of a figure-eight with a marked sheet, so counting them is counting pairs of permutations that join every sheet to every other. Marshall Hall did it in 1949 by sorting every pair by how far one sheet can travel, and the recursion that comes out says the count is almost exactly n times n factorial.
Named alongside it
The objects these essays reach for when they reach for this one.
PermutationCommutatorAbelian groupAlternating groupCovering spaceDihedral groupExhaustive searchNormal subgroupOrbitPlatonic solidsSimple groupStabiliser