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Concept

Stabiliser

The set of symmetries in a group that leave a chosen object exactly where it is. It is a subgroup, and its size multiplied by the size of the object's orbit gives the size of the whole group — the orbit–stabiliser theorem, which is how most counts of symmetric configurations are made.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

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.

algebra · Symmetry groups 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.

algebra · Symmetry groups

Named alongside it

The objects these essays reach for when they reach for this one.

Counting argumentDihedral groupGroup actionLagrange theoremOrbitAlternating groupAutomorphismConjugacy classCosetCyclic groupLabelled treeNecklace

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