Group action
Named by 8 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.
Named alongside it
The objects these essays reach for when they reach for this one.
Cyclic groupCounting argumentDihedral groupLagrange theoremOrbitPermutationSymmetryCosetCounting two waysEquivalenceExhaustive searchInvariant