Symmetry groups — the series

Every relabelling of a 4-gon's corners, and the 8 that are motions. All 24 permutations of the corners drawn one by one, with the 8 that preserve every distance marked; the rest deform the polygon and are not symmetries.

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.

part 1 · algebra

16 colourings in 6 classes. Every way of colouring the corners, with the ones a motion carries to each other placed on the same row; the number of rows is the number of genuinely different colourings.

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.

part 2 · algebra

A subgroup of 2, and the 4 blocks it cuts the group into. The 8 symmetries of a 4-gon, split into 4 blocks by composing every element onto the subgroup {e, r²}. The blocks all have 2 elements and no element is in two of them.

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.

part 3 · algebra

Orbit times stabiliser is 8, on every row. A table of 5 things the 8 symmetries of a 4-gon can move. Each row draws every position the thing can be carried to and lists the motions that leave it where it is; the two counts multiply to 8 on every row.

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.

part 4 · algebra

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.

part 5 · algebra

Which pairs of the square's symmetries commute: 40 of 64. A 8-by-8 grid over the elements of the 8 symmetries of a 4-gon, rows and columns grouped into 5 conjugacy classes, with a filled square wherever the two elements commute: 40 filled squares, and each row's count of filled squares written at its end.

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.

part 6 · algebra