Dihedral group
Named by 8 essays across 2 fields — each of them below, with the objects they name alongside it.
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.
Half a circle against a wall
Lay a fence of fixed length with both ends against a straight wall and the best shape is a half-circle, holding exactly twice what a full circle of the same fence holds. The proof is a mirror: doubled in the wall, any fence becomes a closed curve with twice the length and twice the area, and the closed-curve answer carries over. In a corner the same mirrors give a slice of a circle — until the corner's angle stops dividing a half-turn.
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.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Group actionCounting argumentCyclic groupLagrange theoremCosetOrbitStabiliserSymmetryConjugacy classEquivalencePermutationRegular polygon