Group action
Named by 15 essays across 4 fields — each of them below, with the objects they name alongside it.
Necklaces that prove a theorem
Thread five beads in two colours, thirty-two ways. Two of them are all one colour; the other thirty fall into rings of five. That count, and nothing else, is Fermat's little theorem.
Eight ways to leave a square alone
A square can be picked up and put back so that nothing looks different. There are exactly eight ways to do it, and the number is not asserted here — it is what a search through all twenty-four relabellings of the corners comes back with.
Colourings nobody can tell apart
Sixteen ways to colour four corners in two colours, and only six of them are genuinely different. The count can be got by pooling the sixteen — or by never forming a single class and instead averaging how many colourings each motion leaves untouched.
The blocks a subgroup cuts out
Take any part of a group that is closed under composition, and it slices the whole group into blocks of its own size that do not overlap. Everything Lagrange's theorem says is arithmetic about that picture — and whether the blocks can be multiplied is a separate question with a surprising answer.
The group drawn as a map
A multiplication table says everything about a group and shows nothing; lay the same information out as one dot per element and one arrow per generator, and multiplying becomes walking, distance becomes a word length, and the group acquires a shape.
Sixteen of five hundred and seventy-six
A Latin square is a multiplication table in which every equation has exactly one solution. Ask it to be associative as well and almost every square drops out — sixteen of the five hundred and seventy-six of order four survive, and they are the two groups.
The puzzle that is exactly half solvable
A sliding puzzle sold with two tiles swapped is not a hard puzzle; it is an impossible one, and the proof is a quantity that no slide can change. The same argument, run three times at once, says that one arrangement of a scrambled cube in twelve is reachable.
A covering is a permutation
Describing a covering means saying where each loop sends each sheet, which is a permutation for every generator. So a covering of a wedge of circles is nothing but a homomorphism to a symmetric group, and the subgroup it corresponds to is a stabiliser.
Every third coefficient
Add every third number in the twelfth row of Pascal's triangle and the answer is 1366 — a third of 4096, rounded up. Which way the rounding goes is decided by two arrows of length one in the complex plane, and the same average over the roots of unity counts dice totals, subsets and necklaces.
One residue whose powers are all of them
Fermat's theorem says every order divides p − 1. It does not say that anything has order exactly p − 1, which is a separate and stronger claim — and what forces it is a count of how many numbers share each divisor with p − 1.
How fast the ball fills
Count the elements within r steps of doing nothing. The count grows like a polynomial in some groups and like a power of three in others, the distinction survives every change of generating set, and which polynomial degrees are possible is a theorem nobody expected.
The edge that is as big as the ball
In a lattice the boundary of a large ball is a negligible fraction of it. In a tree it is two thirds of it at every size — and that single ratio, not the group's size, is what decides whether a set can be cut into pieces and reassembled into two copies of itself.
A count that can say zero
The branched count ends on a list of cycle shapes that passes every test and describes no covering. There is an exact formula for how many coverings a list has — a sum over the character table of a symmetric group — and it returns nought without giving any reason why.
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.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Cyclic groupCounting argumentDihedral groupOrbitLagrange theoremInvariantModular arithmeticPermutationCayley graphCosetCounting two waysFree group